The given function is exponential. Meanwhile \(change of base\) 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
\(y = {3}^{x}\)
Writing the above exponential equation as a logarithmic equation, we obtain;
\(log_{3}(y) = x\)
We can apply the change of base formula to obtain,
\(\frac{ log_{e}(y) }{ log_{e}(3) } = x\)
Or
\(\frac{ ln(y) }{ln(3) } = x\)
We can cross multiply to obtain;
\(ln(y) = x ln(3)\)
Taking logarithm of both sides to base e, we obtain;
\({e}^{ ln(y) } = {e}^{x ln(3) }\)
This implies that,
\(y= {e}^{x ln(3) }\)
But we know that,
\(y = {3}^{x}\)
Hence
\({3}^{x} = {e}^{x ln(3)}\)
