Question

How do you calculate \[{{\log }_{7}}\left( 42 \right)\]?

Answer 1

Hint: From the question given, we have been asked to calculate \[{{\log }_{7}}\left( 42 \right)\].We can clearly see that the given expression is a logarithmic expression. We can calculate the given logarithmic expression by using some basic law of logarithms or some basic formulae of logarithms.

Complete step by step answer:
From the question given, we have been given to calculate \[{{\log }_{7}}\left( 42 \right)\]
Here, to solve the given question, we have to apply the base change rule in logarithms.
The formula for base change rule in logarithms is shown below,
Base change rule: \[{{\log }_{b}}a=\dfrac{{{\log }_{c}}a}{{{\log }_{c}}b}\]
We have to apply the above shown base change rule to solve the given question.
By applying the base change rule to the given question, we get \[\Rightarrow {{\log }_{7}}42=\dfrac{{{\log }_{10}}42}{{{\log }_{10}}7}\]
Now, let us split \[42\] into products of \[6\] and \[7\].
By splitting \[42\] into product of \[6\] and \[7\], we get \[\Rightarrow {{\log }_{7}}42=\dfrac{{{\log }_{10}}\left( 7\times 6 \right)}{{{\log }_{10}}7}\]
In logarithms, we have one basic formula, that is \[\log \left( mn \right)=\log m+\log n\]
Apply this basic formula to the above equation. By applying this basic formula to the above equation, we get
\[\Rightarrow {{\log }_{7}}42=\dfrac{{{\log }_{10}}7+{{\log }_{10}}6}{{{\log }_{10}}7}\]
\[\Rightarrow {{\log }_{7}}42=\dfrac{{{\log }_{10}}7}{{{\log }_{10}}7}+\dfrac{{{\log }_{10}}6}{{{\log }_{10}}7}\]
Now, do more simplification to the above equation to get the perfect answer.
By doing further more simplification to the above equation, we get
\[\Rightarrow {{\log }_{7}}42=1+\dfrac{{{\log }_{10}}6}{{{\log }_{10}}7}\]
\[\Rightarrow {{\log }_{7}}42=1+\dfrac{0.7782}{0.8451}\] (substituting values using pocket calculator)
\[\Rightarrow {{\log }_{7}}42=1.9208\]
Hence, the given question is calculated using the basic formulae of logarithms.

Note:
We should be well aware of the basic formulae and basic laws of logarithms. Also, we should be very careful while doing the calculation part of the above question. Also, we should be very careful while applying the basic laws of logarithms to the given question. Also, we should be well known about the usage of the formulae of logarithms. Similarly we have many logarithm formulae like ${{\log }_{a}}b=\dfrac{\log b}{\log a}$ and many more.