Top 39 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. 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

5

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


Question 2

If $$\log_{12}{81}=p$$, then $$3(\dfrac{4-p}{4+p})$$ is equal to


Question 3

If N and x are positive integers such that $$N^{N}$$ = $$2^{160}\ and \ N{^2} + 2^{N}\ $$ is an integral multiple of $$\ 2^{x}$$, then the largest possible x is


Question 4

$$\frac{1}{log_{2}100}-\frac{1}{log_{4}100}+\frac{1}{log_{5}100}-\frac{1}{log_{10}100}+\frac{1}{log_{20}100}-\frac{1}{log_{25}100}+\frac{1}{log_{50}100}$$=?


Question 5

If p$$^{3}$$ = q$$^{4}$$ = r$$^{5}$$ = s$$^{6}$$, then the value of $$log_{s}{(pqr)}$$ is equal to


Question 6

Given that $$x^{2018}y^{2017}=\frac{1}{2}$$, and $$x^{2016}y^{2019}=8$$, then value of $$x^{2}+y^{3}$$ is


Question 7

If $$\log_{2}({5+\log_{3}{a}})=3$$ and $$\log_{5}({4a+12+\log_{2}{b}})=3$$, then a + b is equal to

CAT 2017 Logarithms, Surds and Indices questions

Question 1

Suppose, $$\log_3 x = \log_{12} y = a$$, where $$x, y$$ are positive numbers. If $$G$$ is the geometric mean of x and y, and $$\log_6 G$$ is equal to


Question 2

If x is a real number such that $$\log_{3}5= \log_{5}(2 + x)$$, then which of the following is true?


Question 3

The value of $$\log_{0.008}\sqrt{5}+\log_{\sqrt{3}}81-7$$ is equal to


Question 4

If $$9^{2x-1}-81^{x-1}=1944$$, then $$x$$ is


Question 5

If $$9^{x-\frac{1}{2}}-2^{2x-2}=4^{x}-3^{2x-3}$$, then $$x$$ is


Question 6

If $$log(2^{a}\times3^{b}\times5^{c} )$$is the arithmetic mean of $$log ( 2^{2}\times3^{3}\times5)$$, $$log(2^{6}\times3\times5^{7} )$$, and $$log(2 \times3^{2}\times5^{4} )$$, then a equals

CAT 2006 Logarithms, Surds and Indices questions

Question 1

If $$log_y x = (a*log_z y) = (b*log_x z) = ab$$, then which of the following pairs of values for (a, b) is not possible?


Question 2

If x = -0.5, then which of the following has the smallest value?


Question 3

Which among $$2^{1/2}, 3^{1/3}, 4^{1/4}, 6^{1/6}$$, and $$12^{1/12}$$ is the largest?

CAT 2005 Logarithms, Surds and Indices questions

Question 1

If x >= y and y > 1, then the value of the expression $$log_x (x/y) + log_y (y/x)$$ can never be

CAT 2004 Logarithms, Surds and Indices questions

Question 1

Let $$u = ({\log_2 x})^2 - 6 {\log_2 x} + 12$$ where x is a real number. Then the equation $$x^u = 256$$, has

CAT 2003 Logarithms, Surds and Indices questions

Question 1

If $$log_3 2, log_3 (2^x - 5), log_3 (2^x - 7/2)$$ are in arithmetic progression, then the value of x is equal to

CAT 2002 Logarithms, Surds and Indices questions

Question 1

If $$f(x) = \log \frac{(1+x)}{(1-x)}$$, then f(x) + f(y) is

Show Answer Explanation

CAT 1997 Logarithms, Surds and Indices questions

Question 1

If $$\log_{2}{\log_{7}{(x^2 - x+37)}}$$ = 1, then what could be the value of ‘x’?


Question 2

Which of the following is true?

Show Answer Explanation

CAT 1996 Logarithms, Surds and Indices questions

Question 1

Find the value of $$\frac{1}{1 + \frac{1}{3-\frac{4}{2+\frac{1}{3-\frac{1}{2}}}}}$$ + $$\frac{3}{3 - \frac{4}{3+\frac{1}{2-\frac{1}{2}}}}$$

Show Answer Explanation

CAT 1991 Logarithms, Surds and Indices questions

Question 1

$$2^{73}-2^{72}-2^{71}$$ is the same as

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