First slide

Types of real valued functions XI

Question

If $\log_{k} x \cdot \log_{5} k = \log_{x} 5, k \neq 1, k > 0$ then x is equal to

Moderate

A

k
B

1/5
C

5
D

none of these

Solution

$\begin{array}{ll} \log_{k} x \cdot \log_{5} k = \log_{x} 5 \\ or \frac{\log x}{\log k} \frac{\log k}{\log 5} = \log_{x} 5 \\ or \frac{\log x}{\log 5} = \log_{x} 5 \\ or \log_{5} x = \frac{1}{\log_{5} x} \\ or {(\log_{5} x)}^{2} = 1 \\ or \log_{5} x = \pm 1 \\ \Rightarrow x = 5^{\pm 1} \\ \Rightarrow x = \frac{1}{5}, 5 \end{array}$

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