# CAT Logarithms, Surds and Indices Questions With Video Solutions

CAT Logarithms, Surds and Indices questions are the important questions frequently appearing in the CAT examination. Practising mocks for CAT where you'll get a fair idea of how questions are asked, and type of questions asked in CAT Exam. These questions require a solid understanding of fundamental concepts. To help the aspirants, we have compiled all the questions from this topic that appear in the previous CAT papers, along with the video solutions for every question explained in detail by the CAT toppers. One can download them in a PDF format or take them in a test format. Click on the link below to download the CAT Logarithms, Surds and Indices questions with detailed video solutions PDF.

## CAT Logarithms, Surds And Indices Questions Weightage Over Past 5 Years

 Year Weightage 2023 4 2022 1 2021 3 2020 7 2019 4 2018 7

## CAT Logarithms, Surds and Indices Formulas PDF

Logarithms, surds and indices questions are frequently asked in the previous CAT papers. In order to ace this topic and solve the CAT questions, aspirants must be well-versed in the basic concepts and formulas. To help the aspirants, we have made a PDF which consists of all the formulas, tips and tricks to solve these questions. Every formula in this PDF is very important. Click on the below link to download the CAT Logarithms, Surds and Indices Formulas PDF.

1. Formula: Properties of logarithm

$$\log_{a}{1} = 0$$$$$\log_{a}{xy} = \log_{a}{x}+\log_{a}{y}$$$ $$\log_{a}{b}^{c} = c \log_{a}{b}$$$$${b}^{\log_{b}{x}} = x$$$ $${x}^{\log_{b}{y}} = {y}^{\log_{b}{x}}$$$$${\log_{a}{\sqrt[n]{b}}} = \dfrac{\log_{a}{b}}{n}$$$ $${\log_{a}{b}} = \dfrac{\log_{c}{b}}{\log_{c}{a}}$$$$${\log_{a}{b}}*{\log_{b}{a}}= 1$$$ $$a^m\times\ a^n=a^{m+n}$$$$$\frac{a^m\ \ }{a^n}\ =a^{m-n}$$$ $$\left(a^m\right)^{^n}=a^{m\times\ n}$$$$$\left(a\times\ b\right)^m\ =a^m\times\ b^m$$$ $$a^{-m}=\ \frac{1}{a^m}$$$$$a^{\frac{m}{n}}=\sqrt[\ n]{a^m}$$$

• Logarithms can be used to quickly find the number of digits in an exponent.

## CAT 2023 Logarithms, Surds and Indices questions

#### Question 1

If x is a positive real number such that $$x^8 + \left(\frac{1}{x}\right)^8 = 47$$, then the value of $$x^9 + \left(\frac{1}{x}\right)^9$$ is

#### Question 2

If $$x$$ and $$y$$ are positive real numbers such that $$\log_{x}(x^2 + 12) = 4$$ and $$3 \log_{y} x = 1$$, then $$x + y$$ equals

#### Question 3

If $$\sqrt{5x+9} + \sqrt{5x - 9} = 3(2 + \sqrt{2})$$, then $$\sqrt{10x+9}$$ is equal to

#### Question 4

For some positive real number x, if $$\log_{\sqrt{3}}{(x)}+\frac{\log_{x}{(25)}}{\log_{x}{(0.008)}}=\frac{16}{3}$$, then the value of $$\log_{3}({3x^{2}})$$ is

## CAT 2022 Logarithms, Surds and Indices questions

#### Question 1

If $$(\sqrt{\frac{7}{5}})^{3x-y}=\frac{875}{2401}$$ and $$(\frac{4a}{b})^{6x-y}=(\frac{2a}{b})^{y-6x}$$, for all non-zero real values of a and b, then the value of $$x+y$$ is

## CAT 2021 Logarithms, Surds and Indices questions

#### Question 1

For a real number a, if $$\frac{\log_{15}{a}+\log_{32}{a}}{(\log_{15}{a})(\log_{32}{a})}=4$$ then a must lie in the range

#### Question 2

If $$\log_{2}[3+\log_{3} \left\{4+\log_{4}(x-1) \right\}]-2=0$$ then 4x equals

#### Question 3

If $$5 - \log_{10}\sqrt{1 + x} + 4 \log_{10} \sqrt{1 - x} = \log_{10} \frac{1}{\sqrt{1 - x^2}}$$, then 100x equals

## CAT 2020 Logarithms, Surds and Indices questions

#### Question 1

If Y is a negative number such that $$2^{Y^2({\log_{3}{5})}}=5^{\log_{2}{3}}$$, then Y equals to:

#### Question 2

If $$\log_{a}{30}=A,\log_{a}({\frac{5}{3}})=-B$$ and $$\log_2{a}=\frac{1}{3}$$, then $$\log_3{a}$$ equals

#### Question 3

The value of $$\log_{a}({\frac{a}{b}})+\log_{b}({\frac{b}{a}})$$, for $$1<a\leq b$$ cannot be equal to

#### Question 4

If a,b,c are non-zero and $$14^a=36^b=84^c$$, then $$6b(\frac{1}{c}-\frac{1}{a})$$ is equal to

#### Question 5

If $$x=(4096)^{7+4\sqrt{3}}$$, then which of the following equals to 64?

#### Question 6

If $$\log_{4}{5}=(\log_{4}{y})(\log_{6}{\sqrt{5}})$$, then y equals

#### Question 7

$$\frac{2\times4\times8\times16}{(\log_{2}{4})^{2}(\log_{4}{8})^{3}(\log_{8}{16})^{4}}$$ equals

## CAT 2019 Logarithms, Surds and Indices questions

#### Question 1

If $$(5.55)^x = (0.555)^y = 1000$$, then the value of $$\frac{1}{x} - \frac{1}{y}$$ is

#### Question 2

The real root of the equation $$2^{6x} + 2^{3x + 2} - 21 = 0$$ is

#### Question 3

If m and n are integers such that $$(\surd2)^{19} 3^4 4^2 9^m 8^n = 3^n 16^m (\sqrt[4]{64})$$ then m is

#### Question 4

Let x and y be positive real numbers such that
$$\log_{5}{(x + y)} + \log_{5}{(x - y)} = 3,$$ and $$\log_{2}{y} - \log_{2}{x} = 1 - \log_{2}{3}$$. Then $$xy$$ equals

## CAT 2018 Logarithms, Surds and Indices questions

#### Question 1

If x is a positive quantity such that $$2^{x}=3^{\log_{5}{2}}$$. then x is equal to