Question

Solve the logarithmic-exponential function:$\ln {e^x} = 4$ and find the value of $x$

Answer 1

We are given a natural logarithmic function which has a base of $e$ and we have to evaluate its value. For this we use the conversion of logarithms into exponents because both are inverse entities of each other and then by comparison of indices we will obtain our result.

Complete solution step by step:
Firstly we write the given logarithmic expression
$\ln {e^x} = 4$
Exponent of a number means how many times the number is multiplied by itself i.e.
${p^q} = \underbrace {p \times p \times p \times p......}_{q\,{\text{times}}} = r$
It says $p$ multiplied by itself $q$ times equals to $r$
And logarithms are just opposite to it where the following function
${\log _a}c = b\,{\text{ - - - - - - - - equation(1)}}$
Means that - When $a$ is multiplied by itself $b$ number of times $c$ is obtained.
So we translate this into our equation
$\ln {e^x} = 4$
Here the base of the logarithm is $e$ so we can say by this equation that- $e$ is multiplied with itself 4 times to obtain ${e^x}$ i.e.
$e \times e \times e \times e = {e^x} \\\ \Rightarrow {e^4} = {e^x} \\\$
Here we used the following property of indices
${p^q} = {p^r} \\\ \Rightarrow q = r \\\$
$\Rightarrow x = 4\left( {\because {a^b} = {a^c}, \Rightarrow b = c} \right)$
Hence, we have obtained our result using the definition of natural logarithms.
Additional information:
Exponential function is the Inverse function of a logarithmic function. This means one can be undone or removed by operating the other function on it and vice versa. This would give you a better understanding of it –
${\log _a}c = b \\\ {a^b} = {a^{{{\log }_a}\;c}} = c \\\$
Doing the opposite will give us –
${a^b} = c \\\ {\log _a}c = {\log _a}({a^b}) = b \\\$
The above result could be used in our question directly i.e.
\ln {e^x} = 4 \\\ \Rightarrow {\log _e}\left( {{e^x}} \right) = 4,\left\\{ {\because \ln a = {{\log }_e}a} \right\\} \\\ \Rightarrow x = 4 \\\
This helps us to understand the reason why they are inverse functions with each other providing the same answer using both methods.

Note: To evaluate the value of logarithm, we try to reach from bottom to top of a logarithmic function by using some numbers and operations. Both the methods i.e. translating definition of logarithms or logarithmic property for exponent conversion can be used to solve the problem.