CAT Logarithms, Surds and Indices Questions with Answers

Logarithms, Surds and Indices

The subtopics that fall under logarithms and surds are surds, logarithms in different bases, and finding the number of digits. The following practice questions come with detailed explanations and video solutions. 

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Instructions

For the following questions answer them individually

Question 1

Find the sum of all the values of ‘a’ that satisfy the following equation:

$$2^{\log_7 |a+9|} = \log_7 2401$$

Question 2

'$$p$$' and '$$q$$' are two positive numbers such that $$p>q$$. What is the maximum value of the expression

$$p^{\sqrt{\log _p q}} - q^{\sqrt{\log _q p}} $$

Question 3

a, b, c and d are natural numbers less than 1000 such that $$\log _a b = 7/5$$ and $$\log _c d = 4/3$$. Given that d-b=497, what is a+c?

Question 4

How many values of ‘p’ satisfy the following equation:
$$(\log_5 p)^2 + \log_{5p} (5/p) = 1$$?

Question 5

If $$log_z{x}=\frac{1}{3}$$ and $$log_w{y}=\frac{1}{4}$$ where x, y, z, w are distinct natural numbers such that x, y, z, w are distinct natural numbers such that x<y<z<w, what is the minimum possible value of x+y+z+w?

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Question 6

What is the value of $$\frac{1}{1+a^{y-x}+a^{z-x}} + \frac{1}{1+a^{x-y}+a^{z-y}} + \frac{1}{1+a^{x-z}+a^{y-z}}$$?

Question 7

$$x^{1-x}=y$$ , $$y^{1-y}=z$$ and $$z^{1-z}=x$$; x, y and z are greater than 0. Find the value of xy+yz.

Question 8

The number of digits in $$(2401^{35})_7 $$ is? (Subscript represents the base in which the number is written)

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Question 9

$$ \log _9 X + \log _{27} \sqrt{X} = (\log _{49} 11)/(\log _7 3)$$.

Find X.

Question 10

$$\log _a b = 2;\log _b c = 3/2$$ and $$\log _c d = 4/3$$ How many solutions are possible for a,b,c and d if they are all natural numbers less than 1000?

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Question 11

For every value of 'a', how many values of b satisfy the equation $$log_{b}a+ log_{ab}a^2 + log_{a^{2}b}a^{3} =0$$, when $$a>1$$ and $$a \neq b$$? (Enter -1, if the answer cannot be determined)

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Question 12

Find the number of zeroes before the first non-zero digit to the right of the decimal point in $$1/60^{20}$$.

$$log_{10}2 = 0.3010$$, $$log_{10}3 = 0.4771$$

Question 13

Solve: $$ \log _{10} (25-x) - \log _{2} 4 = \log_{10} x - 2\log_{10} 5 $$

Question 14

Solve: $$\log 325 + 3 \log 4 - \log 455$$

Question 15

For which of the following range of ‘a’ the following inequality holds good:

$$\log_a 4 + \log_{a^3} 8$$ > 1?

Question 16

What is the number of digits in the number $$60^{20}$$. (log 2 =0.3010 , log 3=0.4771) ?

Question 17

If $$\log _a {147} = X$$ and $$\log _a {63} = Y$$, find the value of $$\log _a {441}$$

Question 18

Simplify the surd $$\sqrt{10+\sqrt{75}} + \sqrt{10-\sqrt{75}}$$

Question 19

Find the value of log 0.24242424…, given that log 2 = 0.3010, log 3 = 0.4771, log 5 = 0.6989, log 11 = 1.0413.

Question 20

Which of the following surds is the greatest?

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