Question

Is it true or false? \[{\log _9}59049 = 5\]

Answer 1

Hint: Here we have to use laws of logarithm to get whether the statement is correct or not. Check whether a given number on LHS is a power of 9 or not because in base we have 9. This will help in proceeding with the problem.
Formula used:
\[{\log _b}a = \dfrac{{\log a}}{{\log b}}\]
\[\log \left( {{a^n}} \right) = n\log a\]

Complete step-by-step answer:
Given that,
\[{\log _9}59049 = 5\]
Let’s take its LHS.
\[{\log _9}59049\]
Using first formula from list,
\[ \Rightarrow \dfrac{{\log 59049}}{{\log 9}}\]
Now 59049 is the fifth power of 9.
That is \[{9^5} = 59049\]
So above log can be written as,
\[ \Rightarrow \dfrac{{\log \left( {{9^5}} \right)}}{{\log 9}}\]
Using second formula from list,
\[
   \Rightarrow \dfrac{{5\log 9}}{{\log 9}} \\
   \Rightarrow 5 \\
\]
=RHS
Hence the given statement is true.

Note: We directly cannot judge for the statement. Students should see which rules of logarithm can be used so that the sum can be simplified at its best. Do list out the rules of logarithm.