Quadratic Logarithmic Equations

Quadratic Logarithmic Equations

Edited By Komal Miglani | Updated on Jul 02, 2025 07:35 PM IST

Logarithmic equations are a fundamental aspect of algebra and are widely used in various scientific and engineering fields. Sometimes, logarithmic equations can be transformed into a quadratic form, enabling the use of techniques from quadratic equations to solve them. Logarithmic equations can be expressed in quadratic form, exploring their properties, solution methods, and applications.

This Story also Contains
  1. Logarithmic Equations in Quadratic form
  2. Summary
  3. Solved Examples Based on Logarithmic Equations in Quadratic form
Quadratic Logarithmic Equations
Quadratic Logarithmic Equations

Logarithmic Equations in Quadratic form

A polynomial equation in which the highest degree of a variable term is 2 is called a quadratic equation.

Standard form of a quadratic equation is $a x^2+b x+c=0$

Where a, b, and c are constants (they may be real or imaginary) and called the coefficients of the equation and $a \neq 0$ (a is also called the leading coefficient).

Eg, $-5 x^2-3 x+2=0, x^2=0,(1+i) x^2-3 x+2 i=0$

As the degree of the quadratic polynomial is 2, so it always has 2 roots (number of real roots + number of imaginary roots = 2)

Roots of quadratic equation

The root of the quadratic equation is given by the formula:

$\begin{aligned} & x=\frac{-b \pm \sqrt{D}}{2 a} \\ & \text { or } \\ & x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}\end{aligned}$

Where D is called the discriminant of the quadratic equation, given by $D=b^2-4 a c$

Logarithmic Equations:

Equation of the form $\log _{\mathrm{a}} \mathrm{f}(\mathrm{x})=\mathrm{b}(\mathrm{a}>0, \mathrm{a} \neq 1)$, is known as logarithmic equation.
this is equivalent to the equation $\mathrm{f}(\mathrm{x})=\mathrm{a}^{\mathrm{b}}(\mathrm{f}(\mathrm{x})>0)$

Let us see one example to understand

Suppose given equation is $\log _{\log _4 x} 4=2$
base of $\log$ is greater then 0 and not equal to 1
so, $\log _4 x>0$ and $\log _4 x \neq 1$
$x>1$ and $x \neq 4$
now, using $\log _a f(x)=b \Rightarrow f(x)=a^b$

$
\begin{aligned}
& \Rightarrow 4=2^{\log _4 x} \Rightarrow 2^2=2^{\log _4 x} \\
& \Rightarrow 2=\log _4 x \Rightarrow x=4^2 \\
& x=16
\end{aligned}
$

If the given equation is in the form of $f\left(\log _a x\right)=0$, where $a>0$ and $a$ is not equal to 1 . In this case, put $\log _a x=t$ and solve $f(t)=0$.
And if the given equation is in the form of $f\left(\log _x A\right)=0$, where $A>0$. In this case, put $\log _x A=t$ and solve $f(t)=0$.

For example,

Suppose given equation is $\frac{(\log x)^2-4 \log x^2+16}{2-\log x}=0$ given equation can be written as after substituting $t=\log x$

$
\begin{aligned}
& \Rightarrow \frac{t^2-8 \mathrm{t}+16}{2-\mathrm{t}}=0 \\
& \Rightarrow \frac{(\mathrm{t}-4)(\mathrm{t}-4)}{(2-\mathrm{t})}=0 \\
& \Rightarrow \mathrm{t}=4 \\
& \mathrm{t}=\log \mathrm{x}=4 \\
& \because \log \mathrm{x}=\log _{10} \mathrm{x} \\
& \mathrm{x}=10^4
\end{aligned}
$

Summary

Logarithmic equations in quadratic form present an intriguing intersection of logarithmic and quadratic functions. This approach is not only mathematically elegant but also highly practical, with applications in various fields such as mathematics, economics, physics, and engineering. Understanding and mastering these transformations expand our ability to tackle a broader range of mathematical problems, enhancing both theoretical knowledge and practical problem-solving skills.

Recommended Video Based on Quadratic Logarithmic Equations

Solved Examples Based on Logarithmic Equations in Quadratic form

Example 1: What is the solution of the inequation $\log _{x-3}\left(2\left(x^2-10 x+24\right)\right) \geq \log _{(x-3)}\left(x^2-9\right) ?$?

$\begin{aligned} & \text { 1) } x \in[4, \infty] \\ & \text { 2) } x \in(-\infty,-3) \cup[10+\sqrt{43, \infty} \\ & \text { 3) } x \in[10-\sqrt{43}, \infty] \\ & \text { 4) } x \in[10+\sqrt{43}, \infty]\end{aligned}$

Solution

If the given equation is in the form of$f\left(\log _a x\right)=0$, where $a>0$ and a is not equal to 1.

In this case, put$\log _a x=t$ and solve $f(t)=0$

this inequation is equivalent to:

$
\begin{aligned}
& \left(2\left(x^2-10 x+24\right)\right) \geq\left(x^2-9\right) \\
& x^2-9>0 \\
& x-3>1
\end{aligned}
$

on solving these equation we get

$
x \in[10+\sqrt{43}, \infty)
$

Hence, the answer is the option 4.

Example 2: If for $x \in\left(0, \frac{\pi}{2}\right), \log _{10} \sin x+\log _{10} \cos x=-1$ and $\log _{10}(\sin x+\cos x)=\frac{1}{2}\left(\log _{10} n-1\right), n>0$ then the value of n is equal to:

1) 20

2) 16

3) 9

4) 12

Solution
$
\begin{aligned}
& \mathrm{x} \in\left(0, \frac{\pi}{2}\right) \\
& \log _{10} \sin x+\log _{10} \cos x=-1 \\
& \Rightarrow \quad \log _{10} \sin x \cdot \cos x=-1 \\
& \Rightarrow \quad \sin x \cdot \cos x=\frac{1}{10}
\end{aligned}
$

$\begin{aligned} & \log _{10}(\sin x+\cos x)=\frac{1}{2}\left(\log _{10} n-1\right) \\ & \Rightarrow \quad 2 \log _{10}(\sin x+\cos x)=\left(\log _{10} n-\log _{10}\right) \\ & \Rightarrow \quad(\sin x+\cos x)^2=10^{\left(\log _{10} \frac{n}{10}\right)} \\ & \Rightarrow \quad(\sin x+\cos x)^2=\frac{n}{10}\end{aligned}$

$\begin{aligned} \sin ^2 x+\cos ^2 x+2 \sin x \cdot \cos x & =\frac{n}{10} \\ \Rightarrow 1+\frac{1}{5}=\frac{n}{10} \quad \Rightarrow \quad n & =12\end{aligned}$

Hence, the answer is option 4.

Example 3: Let a complex number $z,|z| \neq 1$ satisfy $\log _{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(|z|-1)^2}\right) \leq 2$ Then, the largest value of |z| is equal to _________.

1) 7

2) 6

3) 5

4) 8

Solution

$\begin{array}{r}\log _{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(|z|-1)^2}\right) \leq 2 \\ \frac{|z|+11}{(|z|-1)^2} \geq \frac{1}{2} \\ 2|z|+22 \geq(|z|-1)^2 \\ 2|z|+22 \geq|z|^2+1-2|z| \\ |z|^2-4|z|-21 \leq 0 \\ (|z|-7)(|z|+3) \leq 0\end{array}$

$|z| \in[-3,7]$

$\therefore \quad$ Largest value of $|z|$ is 7

Hence, the answer is option 1.

Example 4: Solve the equation $2 \log _3 x+\log _3\left(x^2-3\right)=\log _3 0.5+5^{\log _5\left(\log _3 8\right)}$

1) $x=0$
2) $x=-2$
3) $x=2$

4) none of the above

Solution

Using properties of logarithm, this equation can be written as
$
\begin{aligned}
& \log _3 x^2+\log _3\left(x^2-3\right)=\log _3 0.5+\log _3 8 \\
& \log _3\left(x^2 \cdot\left(x^2-3\right)\right)=\log _3(0.5 * 8)
\end{aligned}
$

Now we have same base of $\log$ on both sides, so $\log$ can be removed from both sides

$
\begin{aligned}
& x^2\left(x^2-3\right)=0.5 \cdot 8 \\
& x^2\left(x^2-3\right)=4
\end{aligned}
$

$\begin{aligned} & x^4-3 x^2-4=0 \\ & \text { Let } x^2=t \\ & t^2-3 t-4=0 \\ & (t-4)(t+1)=0 \\ & t=4 \text { or } t=-1 \\ & x^2=4 \text { or } x^2=-1 \\ & x= \pm 2\end{aligned}$

Now check whether x= 2 and x = -2 lie in the domain of the original equation.

For x = -2, the first term in the equation is not defined. So it is rejected.

But for x = 2, all the terms are defined.

So x = 2 is the only answer.

Hence, the answer is the option 3.

Example 5: The number of solutions to the equation $\log _{(x+1)}\left(2 x^2+7 x+5\right)+\log _{(2 x+5)}(x+1)^2-4=0, x>0$ $\begin{aligned} & \log _{(x+1)}\left(2 x^2+7 x+5\right)+\log _{(2 x+5)}(x+1)^2-4=0 \\ \Rightarrow & \log _{(x+1)}((x+1)(2 x+5))+2 \log _{(2 x+5)}(x+1)-4=0\end{aligned}$ is:

1) 1

2) 2

3) 3

4) 4

Solution

$\begin{aligned} & \log _{(x+1)}\left(2 x^2+7 x+5\right)+\log _{(2 x+5)}(x+1)^2-4=0 \\ \Rightarrow & \log _{(x+1)}((x+1)(2 x+5))+2 \log _{(2 x+5)}(x+1)-4=0\end{aligned}$
$
\Rightarrow \log _{(x+1)}(x+1)+\log _{(x+1)}(2 x+5)+2 \log _{(2 x+5)}(x+1)-4=0
$

Let $\log _{(x+1)}(2 x+5)=t \Rightarrow \log _{(2 x+5)}(x+1)=\frac{1}{t}$
$
\Rightarrow \log _{(x+1)}(x+1)+\log _{(x+1)}(2 x+5)+2 \log _{(2 x+5)}(x+1)-4=0
$

Let $\log _{(x+1)}(2 x+5)=t \Rightarrow \log _{(2 x+5)}(x+1)=\frac{1}{t}$

$\begin{aligned} & \Rightarrow \quad 1+t+\frac{2}{t}-4=0 \\ & \Rightarrow \quad t^2-3 t+2=0 \\ & \Rightarrow \quad t=1 \quad \text { or } t=2\end{aligned}$

$\begin{aligned} & \Rightarrow \log _{(x+1)}(2 x+5)=1 \text { or } \log _{(x+1)}(2 x+5)=2 \\ & \Rightarrow 2 x+5=(x+1)^1 \text { or } 2 x+5=(x+1)^2 \\ & \Rightarrow x=-4 \quad \text { or } x^2=4\end{aligned}$

$\Rightarrow x=-4$ or $x=2$ or $x=-2$
Given $x>0 \Rightarrow x=2$

$x=2$ also lies in the domain of all the terms, so it is the answer.

Hence, the answer is the option (1).


Frequently Asked Questions (FAQs)

1. What is a quadratic logarithmic equation?
A quadratic logarithmic equation is an equation that combines both quadratic and logarithmic functions. It typically involves logarithms with a quadratic expression inside, or a quadratic equation with logarithmic terms. These equations require a deep understanding of both quadratic and logarithmic properties to solve.
2. How do you identify a quadratic logarithmic equation?
A quadratic logarithmic equation can be identified by the presence of both logarithmic and quadratic elements. Look for logarithms with quadratic expressions inside (e.g., log(x² + 3x + 2)), or quadratic equations with logarithmic terms (e.g., (log x)² + 3 log x + 2 = 0).
3. What are the key steps to solve a quadratic logarithmic equation?
The key steps are: 1) Simplify the equation using logarithmic properties. 2) If possible, express all logarithms with the same base. 3) Use the change of variable technique to convert it into a quadratic equation. 4) Solve the resulting quadratic equation. 5) Substitute back and solve for the original variable. 6) Check solutions in the original equation and consider domain restrictions.
4. Why can't we always apply logarithms to both sides of a quadratic equation to solve it?
Applying logarithms to both sides of a quadratic equation isn't always possible or helpful because: 1) It only works if both sides are positive. 2) It often complicates the equation rather than simplifying it. 3) It can introduce extraneous solutions that don't satisfy the original equation.
5. How does the domain of a logarithmic function affect solutions to quadratic logarithmic equations?
The domain of a logarithmic function (x > 0 for natural log) restricts the possible solutions. When solving quadratic logarithmic equations, we must check if the solutions satisfy these domain restrictions. Solutions that don't meet these conditions are extraneous and must be discarded.
6. What is the significance of the base in a quadratic logarithmic equation?
The base of the logarithm in a quadratic logarithmic equation is crucial because: 1) It determines the properties and rules we can apply. 2) Equations with different bases may require conversion to a common base before solving. 3) The choice of base can simplify or complicate the solving process.
7. How do you handle a quadratic logarithmic equation with multiple logarithms?
To handle multiple logarithms: 1) Try to express all logarithms with the same base. 2) Use logarithmic properties to combine or separate terms. 3) If possible, use the change of variable technique to transform the equation into a quadratic form. 4) Solve the resulting equation and check solutions in the original equation.
8. What is the change of variable technique in solving quadratic logarithmic equations?
The change of variable technique involves substituting the logarithmic term with a new variable (e.g., let y = log x). This transforms the quadratic logarithmic equation into a standard quadratic equation in terms of the new variable, which can then be solved using familiar quadratic solving methods.
9. Why might a quadratic logarithmic equation have no real solutions?
A quadratic logarithmic equation might have no real solutions because: 1) The domain restrictions of logarithms (argument must be positive) aren't satisfied. 2) The resulting quadratic equation after transformation has no real roots. 3) The solutions to the transformed equation don't satisfy the original logarithmic constraints.
10. How do you verify solutions to a quadratic logarithmic equation?
To verify solutions: 1) Substitute each solution back into the original equation. 2) Check if the solution satisfies domain restrictions (e.g., argument of logarithm must be positive). 3) Ensure the equation balances with the solution. 4) Be aware that the solving process might introduce extraneous solutions that need to be discarded.
11. What's the difference between solving a quadratic equation and a quadratic logarithmic equation?
The main differences are: 1) Quadratic logarithmic equations often require logarithmic manipulation before quadratic solving techniques can be applied. 2) Domain restrictions for logarithms must be considered. 3) The change of variable technique is often necessary for quadratic logarithmic equations. 4) Checking for extraneous solutions is crucial in quadratic logarithmic equations.
12. Can the quadratic formula be directly applied to quadratic logarithmic equations?
The quadratic formula cannot be directly applied to quadratic logarithmic equations. First, the equation must be transformed into a standard quadratic form, usually through logarithmic manipulation and the change of variable technique. Only then can the quadratic formula be applied to the resulting quadratic equation.
13. How do exponential functions relate to quadratic logarithmic equations?
Exponential functions are the inverse of logarithmic functions. In solving quadratic logarithmic equations, we often use this relationship to "undo" the logarithm by applying the corresponding exponential function to both sides of the equation. This step is crucial in the solving process.
14. What role does the natural logarithm (ln) play in quadratic logarithmic equations?
The natural logarithm (ln) often appears in quadratic logarithmic equations. It's particularly useful because: 1) It simplifies many calculations. 2) It's the inverse of the exponential function e^x. 3) Other logarithms can be expressed in terms of ln using the change of base formula. This makes ln a powerful tool in solving these equations.
15. How do you approach a quadratic logarithmic equation where the quadratic term is inside the logarithm?
When the quadratic term is inside the logarithm (e.g., log(ax² + bx + c) = k), approach it by: 1) Applying the exponential function to both sides to "undo" the logarithm. 2) Solving the resulting quadratic equation. 3) Checking solutions against domain restrictions and the original equation.
16. What are common mistakes students make when solving quadratic logarithmic equations?
Common mistakes include: 1) Forgetting to check domain restrictions. 2) Incorrectly applying logarithmic properties. 3) Neglecting to verify solutions in the original equation. 4) Misusing the change of variable technique. 5) Failing to consider the possibility of extraneous solutions. 6) Incorrectly assuming all quadratic solving techniques apply directly to logarithmic equations.
17. How does the complexity of quadratic logarithmic equations compare to regular quadratic equations?
Quadratic logarithmic equations are generally more complex because: 1) They combine two types of functions (quadratic and logarithmic). 2) They often require multiple solving steps and techniques. 3) Domain restrictions add an extra layer of consideration. 4) The potential for extraneous solutions is higher. 5) They often require a deeper understanding of both quadratic and logarithmic properties.
18. Can graphing be used to solve quadratic logarithmic equations?
Yes, graphing can be a useful tool for solving quadratic logarithmic equations. By graphing both sides of the equation separately and finding their intersection points, you can visualize the solutions. However, this method is often used as a supplement to algebraic solving, as it may not always provide exact solutions.
19. What is the significance of the number e in quadratic logarithmic equations?
The number e is significant in quadratic logarithmic equations because: 1) It's the base of natural logarithms (ln). 2) It simplifies many calculations involving exponentials and logarithms. 3) It often appears when solving equations involving ln. 4) Understanding e and its properties is crucial for manipulating and solving complex logarithmic equations.
20. How do you handle a quadratic logarithmic equation with different bases?
To handle different bases: 1) Use the change of base formula to convert all logarithms to a common base (often natural log). 2) Simplify the equation using logarithmic properties. 3) Apply the change of variable technique if necessary. 4) Solve the resulting equation. 5) Check solutions in the original equation and consider domain restrictions.
21. What is the relationship between the solutions of a quadratic logarithmic equation and its graph?
The solutions of a quadratic logarithmic equation correspond to the x-coordinates of the points where the graphs of the left-hand side and right-hand side of the equation intersect. Understanding this relationship can help visualize the number and nature of solutions, even before solving the equation algebraically.
22. How does the presence of constants affect the solving process of quadratic logarithmic equations?
Constants in quadratic logarithmic equations can affect the solving process by: 1) Changing the domain restrictions. 2) Altering the application of logarithmic properties. 3) Influencing the complexity of the resulting quadratic equation after transformation. 4) Potentially introducing or eliminating solutions. Always consider how constants interact with both the quadratic and logarithmic parts of the equation.
23. Can quadratic logarithmic equations have complex solutions?
While logarithms are typically defined for real, positive numbers, quadratic logarithmic equations can lead to complex solutions in certain cases. However, these solutions often don't satisfy the domain restrictions of real logarithms and are usually discarded. It's important to understand when and why complex solutions might arise and how to interpret them in the context of the problem.
24. How do you approach a quadratic logarithmic equation where logarithms appear on both sides?
When logarithms appear on both sides: 1) Try to combine logarithms on each side using logarithmic properties. 2) If possible, isolate logarithms on one side of the equation. 3) Apply the exponential function to both sides to eliminate logarithms. 4) Solve the resulting equation, which may be quadratic. 5) Check solutions against domain restrictions and in the original equation.
25. What role does factoring play in solving quadratic logarithmic equations?
Factoring can be crucial in solving quadratic logarithmic equations, especially after transforming the equation into a quadratic form. It can: 1) Simplify the equation. 2) Help identify solutions more easily. 3) Reveal the nature of solutions (real, repeated, or complex). 4) Aid in understanding the structure of the equation and its solutions.
26. How do you determine the number of solutions a quadratic logarithmic equation will have?
Determining the number of solutions involves: 1) Transforming the equation into a quadratic form. 2) Analyzing the discriminant of the resulting quadratic equation. 3) Considering domain restrictions from logarithms. 4) Checking for extraneous solutions. The final number of solutions may be less than or equal to the number suggested by the quadratic form due to logarithmic constraints.
27. What is the importance of understanding function composition in quadratic logarithmic equations?
Understanding function composition is crucial because quadratic logarithmic equations often involve the composition of logarithmic and quadratic functions. This understanding helps in: 1) Recognizing the structure of the equation. 2) Applying appropriate solving techniques. 3) Interpreting the meaning of solutions. 4) Visualizing the graphical representation of the equation.
28. How do you approach a quadratic logarithmic equation that can't be easily transformed into a standard form?
For equations that resist standard transformation: 1) Look for creative applications of logarithmic properties. 2) Consider using numerical methods or graphing for approximate solutions. 3) Try substitution or other algebraic techniques to simplify the equation. 4) Break the equation into cases based on possible values or ranges. 5) If all else fails, consider advanced techniques like Lambert W function for certain types of equations.
29. What is the significance of the logarithmic identity log_a(x^n) = n*log_a(x) in solving quadratic logarithmic equations?
This identity is crucial because: 1) It allows simplification of logarithms with quadratic expressions inside. 2) It can transform a logarithm of a quadratic into a quadratic of a logarithm. 3) It's often the key step in applying the change of variable technique. 4) It helps in recognizing patterns and structures in complex logarithmic equations.
30. How do you interpret solutions to quadratic logarithmic equations in real-world contexts?
Interpreting solutions involves: 1) Understanding the context of the problem. 2) Considering domain restrictions based on the real-world scenario. 3) Evaluating the reasonableness of solutions. 4) Recognizing that some mathematical solutions may not be practically meaningful. 5) Explaining solutions in terms of the original problem, not just as abstract numbers.
31. What is the role of asymptotes in the graphical representation of quadratic logarithmic equations?
Asymptotes play a crucial role because: 1) Vertical asymptotes occur where the argument of the logarithm approaches zero, indicating domain restrictions. 2) They help visualize the behavior of the function as x approaches certain values. 3) Understanding asymptotes can provide insights into the number and nature of solutions. 4) They're essential for sketching accurate graphs of quadratic logarithmic functions.
32. How does the concept of inverse functions apply to quadratic logarithmic equations?
Inverse functions are crucial in solving quadratic logarithmic equations because: 1) Logarithms and exponentials are inverse functions of each other. 2) This relationship is often used to "undo" logarithms in the solving process. 3) Understanding inverse functions helps in interpreting solutions and checking their validity. 4) It's essential for graphical interpretations, as inverse functions are reflections of each other across y = x.
33. What strategies can be used to simplify complex quadratic logarithmic equations before solving?
Strategies include: 1) Applying logarithmic properties to combine or separate terms. 2) Using the change of base formula to convert to a common logarithmic base. 3) Factoring where possible. 4) Recognizing and using standard logarithmic identities. 5) Grouping similar terms. 6) Substituting variables for complex expressions to simplify the overall structure.
34. How do you handle quadratic logarithmic equations with absolute value terms?
To handle absolute value terms: 1) Consider the equation in two cases: when the expression inside the absolute value is positive or negative. 2) Solve each case separately. 3) Combine solutions, ensuring they satisfy the original equation and domain restrictions. 4) Be aware that absolute value terms can introduce additional complexity and potential solutions.
35. What is the significance of the base of logarithms in quadratic logarithmic equations?
The base is significant because: 1) It determines which logarithmic properties can be applied. 2) Changing the base can simplify or complicate the equation. 3) Some bases (like e or 10) have special properties that can be leveraged. 4) Understanding the relationship between bases is crucial for solving equations with mixed bases.
36. How do you approach a system of quadratic logarithmic equations?
To approach a system: 1) Try to simplify each equation individually using logarithmic properties. 2) Look for opportunities to substitute one equation into another. 3) Consider using elimination or substitution methods after transforming equations. 4) Be aware that the system may have multiple, one, or no solutions due to logarithmic constraints. 5) Check all solutions in both original equations.
37. What role does the product rule of logarithms play in solving quadratic logarithmic equations?
The product rule (log_a(xy) = log_a(x) + log_a(y)) is crucial because: 1) It allows splitting complex logarithmic expressions. 2) It can help in simplifying equations by separating quadratic terms. 3) It's often used in conjunction with the power rule to manipulate equations into solvable forms. 4) Understanding when and how to apply it is key to efficient problem-solving.
38. How do you determine if a quadratic logarithmic equation will lead to a transcendental equation?
A quadratic logarithmic equation may lead to a transcendental equation if: 1) After simplification, it can't be expressed as a polynomial equation. 2) It involves logarithms that can't be isolated or eliminated through algebraic means. 3) It requires numerical methods or special functions (like Lambert W) to solve. Recognizing this early can guide the choice of solving strategy.
39. What is the importance of understanding the range of logarithmic functions in solving quadratic logarithmic equations?
Understanding the range is important because: 1) It helps in identifying possible solutions. 2) It aids in recognizing impossible equations (e.g., log(x) = -5 has no real solutions). 3) It's crucial for graphical interpretations and solution verification. 4) It helps in understanding the behavior of the equation for different input values.
40. How do you approach a quadratic logarithmic equation where the variable appears both inside and outside the logarithm?
For such equations: 1) Try to isolate terms with the variable outside the logarithm.

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