Question

How do you express \[{{\log }_{7}}5\] in terms of common logs?

Answer 1

Hint: Try to express the given expression in terms of common logs first by change of base rule i.e. ${{\log }_{b}}a=\dfrac{\log a}{\log b}$ where the base of log will be ‘10’ in both the numerator and the denominator. Then put the values of $\log 5$ and $\log 7$ to obtain the exact result.

Complete step-by-step solution:
Common log: the common logarithm is the logarithm with base ‘10’. It is also known as the decimal logarithm. Logs of base ‘10’ are usually written as $\log x$ instead of ${{\log }_{10}}x$.
The change of base rule: We can change the base of any logarithm using the formula ${{\log }_{b}}a=\dfrac{\log a}{\log b}$ (in both numerator and denominator the base of log is taken as common base of log i.e. ‘10’)
Considering our expression, \[{{\log }_{7}}5\]
By comparing, we get
a=5 and b=7
So, \[{{\log }_{7}}5\] can be written as
\[{{\log }_{7}}5=\dfrac{\log 5}{\log 7}\] (Here also the base of log is ‘10’ in both numerator and denominator)
Putting the values of $\log 5$ and $\log 7$, we get
${{\log }_{7}}5=\dfrac{\log 5}{\log 7}=\dfrac{0.6989}{0.8450}=0.8271$
This is the required solution of the given question.

Note: In common logs the base of log is always ‘10’. \[{{\log }_{7}}5\] can be written as $\dfrac{{{\log }_{10}}5}{{{\log }_{10}}7}$ by taking common base as ‘10’. But as ${{\log }_{10}}5=\log 5$ and ${{\log }_{10}}7=\log 7$, so it is written as \[{{\log }_{7}}5=\dfrac{\log 5}{\log 7}\]. Some basic logarithmic values should be remembered for faster and accurate calculations.