Question

evaluate the following logarithmic expression without the use of a calculator. write your answer as a fraction reduced to lowest terms. \\(\log_{3}\left(\sqrt5{9}\
ight)\\) answer how to enter your answer (opens in new window)

Explanation:

Step1: Rewrite the radical as an exponent

Recall that $\sqrt[5]{9} = 9^{\frac{1}{5}}$. Also, $9 = 3^2$, so we can substitute that in: $9^{\frac{1}{5}}=(3^2)^{\frac{1}{5}}$.
Using the power - of - a - power rule $(a^m)^n=a^{m\times n}$, we get $(3^2)^{\frac{1}{5}} = 3^{\frac{2}{5}}$. So the original expression $\log_{3}(\sqrt[5]{9})$ becomes $\log_{3}(3^{\frac{2}{5}})$.

Step2: Use the logarithmic property

Recall the logarithmic property $\log_{a}(a^x)=x$, where $a>0,a
eq1$ and $x$ is a real number.
In the expression $\log_{3}(3^{\frac{2}{5}})$, we have $a = 3$ and $x=\frac{2}{5}$. By applying the property $\log_{a}(a^x)=x$, we find that $\log_{3}(3^{\frac{2}{5}})=\frac{2}{5}$.

Answer:

$\frac{2}{5}$