Question

Given the exponential equation 2x = 8, what is the logarithmic form of the equation in base 10?

Answer 1

The logarithmic form is $\bold{\frac{\log _{10}(8)}{\log _{10}(2)}}$

Given:

The exponential equation, $2^{x}=8$

To find:  

The logarithmic form in base 10

Solution:

Let $2^{x}=8 \rightarrow$ equation (1)

Multiply by log to the base 10 on both sides in equation (1), we get

$\log _{10}\left(2^{x}\right)=\log _{10}(8)$

Bring x on left side of $\log _{10}$,then the equation becomes,

 $\begin{array}{l}{x \times \log _{10}(2)=\log _{10}(8)} \\\\ {x=\log _{10}(8) \times \frac{1}{\log _{10}(2)}} \\\\ {x=\frac{\log _{10}(8)}{\log _{10}(2)}}\end{array}$

Answer 2

Answer:

$i) logarithmic\:from :x=log_{2}8\\ii)x = \frac{ log_{10}8 }{log_{10}2}$

Step-by-step explanation:

$We \:know \:that :\\If \:a^{x}=N\:\implies log_{a}N = x$

$Given \: Exponential\:form\:2^{x}=8\\Logarithmic \: form \: log_{2}8=x$

$logarithmic \:form \: of \:\\base \:10 is : \\x= log_{2}8\implies x =\frac{ log_{10}8 }{log_{10}2}$

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