On solving the equation: ₁/2}[_t(₄ 32)]= 2 we get the value of 't' as

Correct Answer is : 62516625/1616625

Correct Answer is : 62516625/1616625

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Logarithms

Question : 51 of 85

Marks: +1, -0

\log_{\frac{1}{2}}[\log_t(\log_4 32)]=

2 we get the value of 't' as

Solution:

Concept:

Use change of base and logarithmic properties to solve stepwise.

Explanation:

First, compute

\log_4 32

Since

4 = 2^2

and

32 = 2^5

, we have

\log_4 32 = \frac{5}{2}

So the equation becomes

\log_{\frac{1}{2}}\left[\log_t\left(\frac{5}{2}\right)\right] = 2

Let

x = \log_t\left(\frac{5}{2}\right)

. Then

\log_{\frac{1}{2}} x = 2

implies

x = \left(\frac{1}{2}\right)^2 = \frac{1}{4}

Thus

\log_t\left(\frac{5}{2}\right) = \frac{1}{4}

. Converting to exponential form:

t^{1/4} = \frac{5}{2}

Raise both sides to the 4th power:

t = \left(\frac{5}{2}\right)^4 = \frac{625}{16}

Answer:

t = \frac{625}{16}

which corresponds to option C.

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