Answer :

syed514
3^x : write in terms of base e would 

3x=e^ln3^x

 =e^xln3
kudzordzifrancis
The given function is exponential. Meanwhile [tex]<b>change of base</b>[/tex] has something to do with logarithms. So we convert the function to logarithm function, change to the required base and convert back to exponential function.

Let

[tex]y = {3}^{x} [/tex]

Writing the above exponential equation as a logarithmic equation, we obtain;

[tex] log_{3}(y) = x[/tex]

We can apply the change of base formula to obtain,

[tex] \frac{ log_{e}(y) }{ log_{e}(3) } = x[/tex]

Or

[tex] \frac{ ln(y) }{ln(3) } = x[/tex]

We can cross multiply to obtain;

[tex] ln(y) = x ln(3) [/tex]

Taking logarithm of both sides to base e, we obtain;

[tex] {e}^{ ln(y) } = {e}^{x ln(3) } [/tex]

This implies that,

[tex]y= {e}^{x ln(3) } [/tex]
But we know that,
[tex]y = {3}^{x} [/tex]
Hence
[tex] {3}^{x} = {e}^{x ln(3)} [/tex]

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