Ratio, Proportion, Indices, LogarithmsPYQ Sept 25Question 4400 of 220

The value of $\displaystyle x$ in $\displaystyle \log_x(4) + \log_x(16) + \log_x(64) = 12$ is

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Correct Answer

✅ Option b — 2

Given equation:

\log_x(4) + \log_x(16) + \log_x(64) = 12

We know that:

4 = 2^2, \quad 16 = 2^4, \quad 64 = 2^6

Using the property of logarithms

\displaystyle \log_b(a^k) = k \log_b(a)

, we can rewrite the terms as:

\log_x(2^2) + \log_x(2^4) + \log_x(2^6) = 12

2\log_x(2) + 4\log_x(2) + 6\log_x(2) = 12

Combining the like terms on the left-hand side:

(2 + 4 + 6) \log_x(2) = 12

12 \log_x(2) = 12

Dividing both sides by 12:

\log_x(2) = 1

By the definition of logarithms,

\displaystyle \log_b(a) = c

implies

\displaystyle b^c = a

. Therefore:

x^1 = 2 \implies x = 2

Hence, **Option B** is the correct answer.

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