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 If \[\log 625 = k\log 5\], then the value of $k$ is……
\[
  {\text{A}}{\text{.5}} \\
  {\text{B}}{\text{.4}} \\
  {\text{C}}{\text{.3}} \\
  {\text{D}}{\text{.2}} \
\]

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Last updated date: 13th Jul 2024
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Answer
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Hint – The Problem is given in the term of logarithm so we Use the logarithm formulas i.e \[\log {a^b} = b\log a\].
Given,
\[\;log625 = klog5\]
We know,
\[625 = {5^4},\,\,\,\,\,\,\,\,\,\,...({\text{i}})\]
We also know,
\[\log {a^b} = b\log a\,\,\,\,\,\,\,\,\,\,\,\,\,...({\text{ii}})\]
From \[{\text{(i) & (ii)}}\] the given equation can be
\[
  \log {5^4} = 4\log 5 = k\log 5 \\
  4\log 5 = k\log 5\,\,\,...({\text{iii}}) \\
  k = 5\,\,\,\,\,\,({\text{from iii}}) \\
\]
Hence the value of $k$ is $5$.
Note – To solve these types of problems of logarithms, we must know the basic formula of log and we must also know the way to use it, like here we have first converted the given equation such that identity can be used in it. Here we have used the formula \[\log {a^b} = b\log a\,\,\].