Question

In the following exercises, solve each logarithmic equation.
$\ln e^{3x}=24$.

Short answer

The solution is$x=8$.

Step-by-step solution

Step 1. Given and explanation.

We have $\log e^{3x}=24$.

We know that if there is an expression as ${\log }_e\left(x\right)=y$, it means that $e^y=\left(x\right)$if we need to convert the logarithmic expression into exponential form.

Then simplifying the equation will give us the answer.

Step 2. Converting into exponential form.

We have $\ln e^{3x}=24$.

This can also be written as ${\log }_e{\left(e\right)}^{3x}=24$.

Converting into exponential forms gives us,

$e^{24}=e^3$.

Step 3. Solving for x.

We now have $e^{24}=e^{3x}$.

Through this we can say that $3x=24$.

Dividing both sides by $3$gives us,

$$\begin{gathered} \frac{3x}{3}=\frac{24}{3} \\ x=8 \end{gathered}$$
Thus we get the solution as $x=8$.

Step 4. Checking the solution

We have $x=8$.

Substituting this in original equation,

$$\begin{gathered} \ln e^{3\times 8}=24 \\ {\log }_e{\left(e\right)}^{24}=24 \\ {e}^{24}=e^{24} \end{gathered}$$
Hence verified.