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Logarithm Question

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Evaluate $$log_{\sqrt[3]{5}}125$$

Mar 22, 2018

#1
+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

Mar 22, 2018
#2
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
+22152
0

Logarithm Question

Evaluate

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

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

Formula:

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

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

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Mar 23, 2018