Question

How do you rewrite \[{\log _5}x\] as a ratio of common logs and natural logs?

Answer 1

Hint: First we are going to assume the above given expression as some variable ‘\[a\]’ then by using formula \[{\log _e}x = a\], it can be converted as \[x = {e^a}\]. Now, we try to convert the equation in common logs and natural logs one by one by taking log both sides as follows.

Formula used:
If \[{\log _e}x = a\] then it can be written as \[x = {e^a}\]
\[{\log _e}{x^a} = a{\log _e}x\]

Complete step by step solution:
Let the given expression in problem be equal to ‘\[a\]’ and write it as follows,
\[ \Rightarrow {\log _5}x = a\] \[\]
By using above formula, we can write the above given equation as following,
\[ \Rightarrow {\log _5}x = a\]
\[ \Rightarrow x = {5^a}\]
Taking both sides logs with base 10, we get
\[ \Rightarrow {\log _{10}}x = {\log _{10}}{5^a}\]
Now it can also be written as following,
\[ \Rightarrow {\log _{10}}x = a{\log _{10}}5\]
Expanding above equation as following,
\[ \Rightarrow {\log _{10}}x = a \times {\log _{10}}5\]
Taking \[{\log _{10}}5\]to the left-hand side and write it as following,
\[ \Rightarrow \dfrac{{{{\log }_{10}}x}}{{{{\log }_{10}}5}} = a\]
Above equation can also be written as,
\[ \Rightarrow a = \dfrac{{{{\log }_{10}}x}}{{{{\log }_{10}}5}}\]
Finally, we got the required ratio of common logs that is with base 10.
\[{\log _5}x\]\[ = \dfrac{{{{\log }_{10}}x}}{{{{\log }_{10}}5}}\] or \[{\log _5}x\]\[ = \dfrac{{\log x}}{{\log 5}}\]
Now for finding the ratio of natural logs, we are repeating all above steps with only a difference of base.
Let the given expression in problem be equal to ‘\[a\]’and write it as follows,
\[ \Rightarrow {\log _5}x = a\] \[\]
By using above formula, we can write the above given equation as following,
\[ \Rightarrow {\log _5}x = a\]
\[ \Rightarrow x = {5^a}\]
Taking both sides logs with base \[e\], we get
\[ \Rightarrow {\log _e}x = {\log _e}{5^a}\]
Now it can also be written as following,
\[ \Rightarrow {\log _e}x = a{\log _e}5\]
Expanding above equation as following,
\[ \Rightarrow {\log _e}x = a \times {\log _e}5\]
Taking \[{\log _e}5\]to the left-hand side and write it as following,
\[ \Rightarrow \dfrac{{{{\log }_e}x}}{{{{\log }_e}5}} = a\]
Above equation can also be written as,
\[ \Rightarrow a = \dfrac{{{{\log }_e}x}}{{{{\log }_e}5}}\]
Finally, we got the required ratio of natural logs that is with base \[e\].
\[{\log _5}x\]\[ = \dfrac{{{{\log }_e}x}}{{{{\log }_e}5}}\] or \[{\log _5}x\]\[ = \dfrac{{\ln x}}{{\ln 5}}\]

Note:
In the above given problem first, we have to understand what is common logs and what is natural logs. Differences between both must be understood so that we can easily solve the given problem. We must have basic knowledge of logs for example what is the base of logs, difference between base 10 and base \[e\]. logs with base 10 are called common logs and logs with base \[e\] are called natural logs.