Question

How do you convert \[{\log _3}29\] to a natural logarithm?

Answer 1

Hint: In the given question, we have been a logarithmic function. We have been given the base and the argument of the function. We have to convert the given log function to the natural log function. The natural log is the one that has a base of “\[e\]”, the Euler’s constant. We are going to have to apply some basic properties of the logarithm function.

Formula Used:
We are going to use the log change of base formula:
\[{\log _a}b = \dfrac{{{{\log }_x}b}}{{{{\log }_x}a}}\]

Complete step by step solution:
We have to convert \[{\log _3}29\] to a natural logarithm.
For doing that, we are going to use the log change of base formula:
\[{\log _a}b = \dfrac{{{{\log }_x}b}}{{{{\log }_x}a}}\]
So, putting \[a = 3\], \[b = 29\] and \[x = e\], we have,
\[{\log _3}29 = \dfrac{{{{\log }_e}29}}{{{{\log }_e}3}}\]
We know, \[{\log _e}\] is standardly written as \[\ln \], thus,

\[{\log _3}29 = \dfrac{{\ln 29}}{{\ln 3}}\]

Additional Information:
The \[\log \] function has other basic properties too:
\[{\log _x}{x^n} = n\]
\[{\log _a}b = \dfrac{1}{{{{\log }_b}a}}\]
\[{\log _b}a = n \Rightarrow {b^n} = a\]

Note:
In the given question, we were given a logarithmic function with the base of a natural number. We had to convert the given base to the base of Euler’s constant “\[e\]”. We did that by using the log change of the base formula. Hence, it is really important that we know the formulae and where, when, and how to use them.