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

 Mar 22, 2018
 #1
avatar+100456 
+1

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

 

 

cool cool cool

 Mar 22, 2018
 #2
avatar
0

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

 

 

so we have( 51/3) = 125 ( = 53 )

 

Therefore x = 9

 Mar 22, 2018
 #3
avatar+22152 
0

Logarithm Question

Evaluate

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

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

 

Formula:

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

 

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

.
 Mar 23, 2018

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