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

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 2

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 3

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

Question 4

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

Find X.

Question 5

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 6

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

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Question 7

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

Question 8

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

Question 9

What is the range of ‘a’ for which the following inequality holds good:

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

Question 10

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

Question 11

'$$p$$' 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 12

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 13

Which of the following surds is the greatest?

Question 14

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

Question 15

$$\log _a b = 2;\log _b c = 3/2$$ 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?

Question 16

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

Question 17

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 18

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

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

Question 19

How many values of ‘p’ satisfy the following equation:

$$(\log_5 p)^2 + \log_{5p} (5/p) = 1$$?

Question 20

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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