How do you evaluate e^(In8)?

Jul 5, 2016

${e}^{\ln} \left(8\right) = 8$

Explanation:

Assuming the intended question is to evaluate ${e}^{\ln} \left(8\right)$:

The base $b$ logarithm is the inverse of an exponential function with base $b$. Specifically, it is the value to which $b$ must be raised to obtain the argument of the function.

${\log}_{b} \left(x\right) = y \iff x = {b}^{y}$

Note, then, that by definition: ${a}^{{\log}_{a} \left(x\right)} = {a}^{y} = x$

(Note that it should make intuitive sense as to why ${a}^{{\log}_{a} \left(x\right)} = x$, as ${\log}_{a} \left(x\right)$ is the power to which we would need to raise $a$ to obtain $x$, and we are raising $a$ to that power)

The natural logarithm, or $\ln$, is the logarithm with the base $e$. With that, we have

${e}^{\ln} \left(8\right) = {e}^{{\log}_{e} \left(8\right)} = 8$