CAT 2026 Logarithms, Surds and Indices: Formulas, Questions and Shortcuts

CAT 2026 Logarithms, Surds and Indices: Important Formulas

Revise these rules before attempting CAT algebra questions. Correctly applying the basic identities can make many multi-step expressions easier to solve.

Logarithm Formulas for CAT

For real logarithms, the base must satisfy a>0 and a1, while the argument must be positive.

Indices and Exponents Rules

Surds Formulas and Identities

CAT Logarithm Questions From Previous Year Papers

The CAT previous year question papers from 2021-2025 include logarithm questions that combine log identities with algebraic equations. The following solved examples demonstrate how to simplify the conditions before calculating the final answer.

Question 1: CAT 2023 Slot 1 - Finding the Values of (x) and (y)

If x and y are positive real numbers such that

x(x2+12)=4

and

3yx=1,

find x+y.

Solution:

From the first equation:

x(x2+12)=4

Convert the logarithmic equation into exponential form:

x2+12=x4

Rearrange:

x4-x2-12=0

Factorise:

(x2-4)(x2+3)=0

Since x is positive, x=2.

Now use the second equation:

3yx=1

yx=13

Converting to exponential form:

y1/3=x

Therefore,

y=x3=23=8

Hence,

x+y=2+8=10

Answer: 10 (Option D).

Shortcut: Convert a logarithmic equation into exponential form as early as possible. Here, it turns the first equation into a simple polynomial equation.

Question 2: CAT 2021 Slot 2 - Nested Logarithms

The CAT 2021 Slot 2 paper includes a question involving nested logarithms:

23+34+4(x-1)-2=0

How to approach it:

First, isolate the outer logarithm:

23+34+4(x-1)=2

Convert it to exponential form:

3+34+4(x-1)=4

Then isolate the remaining logarithm:

4(x-1)=1-34

At this point, use the definition of a logarithm to express x-1 in exponential form, and check the domain condition x>1.

CAT takeaway: For nested logarithms, simplify from the outside inward. Do not combine logarithms with different bases using the product or quotient rules.

CAT Surds Questions: Solved PYQ

Question 3: CAT 2023 Slot 1 - Simplifying a Surd Expression

If

5x+9+5x-9=3(2+2),

find 10x+9.

Solution:

Let

A=5x+9 and B=5x-9.

The given equation becomes:

A+B=6+32

Also,

A2-B2=(5x+9)-(5x-9)=18

Using the difference-of-squares identity:

(A-B)(A+B)=18

Therefore,

A-B=186+32

Rationalise the denominator:

A-B=3(2-2)

We now have:

A+B=6+32

A-B=6-32

Adding both equations:

2A=12A=6

Thus,

5x+9=6

5x+9=36

x=275

Now calculate the required expression:

10x+9=10275+9

=54+9=63=37

Answer: 37 (Option D).

Shortcut: When an equation contains the sum of two square roots, check whether the difference of their squares is simple. Combining the sum and difference can eliminate the need to square the original equation twice.

CAT Indices and Exponents: Practice With PYQ Patterns

Exponent questions can involve large powers, equations with different bases or expressions that become simpler after rewriting the terms.

Question 4: CAT 2023 Slot 2 - Exponential Equation

The CAT 2023 Slot 2 paper includes an equation involving powers of 2 and 4:

24x-22x+16+22x+30=0

Method:

When an equation contains powers of the same base, rewrite each term using a common exponent wherever possible. For example, set t=22x if the equation can be expressed as a quadratic in that quantity.

This substitution can reduce a complicated exponential equation to an ordinary polynomial equation. After solving for t, retain only values consistent with the fact that t is a positive power of 2.

Strengthen your CAT 2026 Quantitative Aptitude preparation with a free PDF containing five years of CAT Quant previous-year questions from 2021 to 2025.

Download the CAT Quant PYQ PDF (2021-2025)

CAT 2026 Logarithms, Surds and Indices: Speed Tricks

Use these approaches to reduce unnecessary calculations in CAT Quantitative Aptitude.

1. Convert logs into exponential form: If ax=b, immediately write x=ab when it simplifies the equation.

2. Use the change-of-base formula: When logarithms have different bases, rewrite them using a common base rather than combining them directly.

3. Check the domain first: For ax, ensure x>0, a>0 and a1.

4. Look for perfect powers: Rewrite numbers such as 8, 16, 27 and 32 as powers of smaller bases to compare exponents.

5. Rationalise strategically: Multiply by the conjugate when a surd expression has a denominator or when the difference of two square roots is useful.

6. Avoid unnecessary squaring: Squaring an equation containing surds can introduce extra solutions. Substitute the final values into the original equation to verify them.

Shortcuts to Solve Logarithms, Surds and Indices Questions Faster

Look for these patterns for surds and indices before starting lengthy calculations, as the right observation can turn a complicated-looking expression into a simple equation.

These shortcuts work best when you first identify the question type-logarithmic equation, surd simplification or comparison of powers-and then choose the quickest valid method.

How to Prepare Logarithms, Surds and Indices for CAT 2026

• Learn the rules first: Memorise the logarithm identities and exponent laws before moving to lengthy questions.

• Practise algebraic manipulation: Work on equations involving unknown bases, nested logarithms and fractional exponents.

• Maintain an error log: Record mistakes involving negative exponents, invalid logarithm domains, surd simplification and incorrect use of log identities.

• Practise under time limits: Once the concepts are clear, solve mixed questions without referring to formulas.

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