Write 1 - 2\log_{7}x as a single logarithm.

Question:

Write {eq}1 - 2\log_{7}x {/eq} as a single logarithm.

Properties Of Logarithms:

There is a set of properties of logarithms which are either used to compress a set of expressions of logarithms into a single logarithm or to expand a single logarithm into a set of expressions of logarithms. A few of them are:

\begin{align} &(i) \,\,\log_a a=1 &(ii) \,\,\log_a x^m = m \log_a x \\ &(iii) \,\, \log _a (mn)=\log _a m - \log_a n \end{align}

The given expression is:

\begin{align} 1 - 2\log_{7}x &= \log_7 7 - 2 \log_7 x & (\because \log_a a=1) \\ &= \log_7 7- \log_7 x^2 & (\because m \log_a x= \log x^m) \\ &= \boxed{\mathbf{ \log_7 \left( \dfrac{7}{x^2} \right)}} & (\because \log _a (mn)=\log _a m - \log_a n ) \end{align}