Logarithm Question

We  can write this in exponential form

[5^(1/3) ] ^x  =  125

(5)^(x/3)  = (5)^3

Since the bases are the same.....we can solve for the exponents

x/3  = 3         multiply both sides by 3

x  =

9        and that's the evaluation for the original expression

Or you can use the definition of a logarithm ie if log_a b = x then a^x = b

so we have( 5^1/3)^x  = 125 ( = 5^{3 )}

Therefore x = 9

Logarithm Question

Evaluate

\(\huge{ \log_{\sqrt[3]{5}}125 } \)

\huge{ \log_{\sqrt[3]{5}}125 }

Formula:

\(\begin{array}{|rcll|} \hline \log_b(x) &=& \dfrac{\log_c(x)}{\log_c(b)} \\ \hline \end{array} \)

\(\begin{array}{|rcll|} \hline && \log_{\sqrt[3]{5}}125 \\\\ &=& \dfrac{ \ln(125) } { \ln(\sqrt[3]{5}) } \\\\ &=& \dfrac{ \ln(5^3) } { \ln(5^{\frac13}) } \\\\ &=& \dfrac{ 3\cdot \ln(5) } {\frac13\cdot \ln(5) } \\\\ &=& \dfrac{ 3 } {\frac13 } \\\\ &=& 3\cdot \frac31 \\\\ &=& 9 \\ \hline \end{array}\)