Question

How do you evaluate ${{e}^{\ln 8}}$ ?

Answer 1

Hint: To solve this question we have to find the value of the given expression, which can be calculated and evaluated by using the basic rules and laws of exponents and logarithms. To find the value of the expression apply the exponent of the log rule, that is ${{b}^{{{\log }_{b}}\left( k \right)}}=k$ .

Complete step-by-step answer:
Given the expression: ${{e}^{\ln 8}}$
A logarithm can be defined as the opposite of a power or exponent, which means that we do take the logarithm of a number we undo the exponentiation on that number.
We have to evaluate the given expression by using the basic logarithmic rules.
We know that the base logarithm $b$ can be represented as the inverse of an exponential function with the same base, that is $b$ .
Also, it can be said that it is the value to which the base $b$ must be raised to obtain the argument of the given function.
Now, from the exponent of log rule ${{b}^{{{\log }_{b}}\left( k \right)}}=k$ and by assuming that in the given question the natural logarithm with base $e$ is taken and comparing the two expressions,
We can evaluate as,
$\Rightarrow {{e}^{\ln 8}}={{e}^{{{\log }_{e}}8}}=8$
Hence, on evaluating the given expression ${{e}^{\ln 8}}$ we get the value as ${{e}^{\ln 8}}=8$ .

Note: While applying the logarithmic rules or laws, one must also know the exponential laws as well which go hand-in-hand with the logarithmic rules. Also, remember that to apply the exponent of the logarithm rule the base should be the same, be it the natural base $e$ or any other base which will be mentioned in the expression.
In short, we can say that the exponent of the logarithm rule states that when raising the logarithm of a given number to its base we get the number itself as the answer.