Question

evaluate the following logarithmic expression without the use of a calculator. write your answer as a fraction reduced to lowest terms. \\(\log_{7}\left(\sqrt3{49}\
ight)\\) answer how to enter your answer (opens in new window)

Explanation:

Step1: Rewrite the radical as an exponent

Recall that $\sqrt[3]{49} = 49^{\frac{1}{3}}$. Also, $49 = 7^2$, so we can substitute that in: $49^{\frac{1}{3}}=(7^2)^{\frac{1}{3}}$.
Using the power - of - a - power rule $(a^m)^n=a^{m\times n}$, we get $(7^2)^{\frac{1}{3}} = 7^{\frac{2}{3}}$. So the expression $\log_{7}(\sqrt[3]{49})$ becomes $\log_{7}(7^{\frac{2}{3}})$.

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 our case, $a = 7$ and $x=\frac{2}{3}$. So $\log_{7}(7^{\frac{2}{3}})=\frac{2}{3}$.

Answer:

$\frac{2}{3}$