If 3x × b5 logb x = 192 then the value of x is

Option c: 2. Solution: Given equation: 3x × b5 logb x = 192 Using the logarithmic property n loga m = loga (mn) : 5 logb x = logb (x5) b5 logb x = blogb (x5) Using the identity blogb y = y : blogb (x5) = x5 Now substitute this back into the original equation: 3x × x5 = 192 3x6 = 192 Divide both sides by 3 : x6 = 192/3 x6 = 64 Express 64 as a power of 2 : 64 = 26 x6 = 26 x = 2 Hence, Option C is the correct answer.

Ratio, Proportion, Indices, LogarithmsPYQ Jan 26Question 4248 of 220

All Questions

If $\displaystyle 3x \times b^5 \log_b x = 192$ then the value of x is

Options

\displaystyle 2\sqrt{2}

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Correct Answer

✅ Option c — 2

All Options:

A8
B4
C2
D $\displaystyle 2\sqrt{2}$

Detailed Solution & Explanation

Given equation:

3x \times b^{5 \log_b x} = 192

Using the logarithmic property

\displaystyle n \log_a m = \log_a (m^n)

5 \log_b x = \log_b (x^5)

b^{5 \log_b x} = b^{\log_b (x^5)}

Using the identity

\displaystyle b^{\log_b y} = y

b^{\log_b (x^5)} = x^5

Now substitute this back into the original equation:

3x \times x^5 = 192

3x^6 = 192

Divide both sides by

\displaystyle 3

x^6 = \frac{192}{3}

x^6 = 64

Express

\displaystyle 64

as a power of

\displaystyle 2

64 = 2^6

x^6 = 2^6 \implies x = 2

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

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If 3x×b5log⁡bx=192\displaystyle 3x \times b^5 \log_b x = 1923x×b5logb​x=192 then the value of x is