Question

Solve the following exponential equation. Exact solutions only. Remember to check your solution(s).



e
x
−
2
=
12
e
x−2
=12

Answer 1

To solve the exponential equation \( e^{x} - 2 = 12 \), we start by isolating the exponential term:

\[
e^{x} - 2 = 12
\]
Adding 2 to both sides gives:

\[
e^{x} = 12 + 2
\]
\[
e^{x} = 14
\]

Next, we take the natural logarithm (ln) of both sides to solve for \( x \):

\[
\ln(e^{x}) = \ln(14)
\]

Using the property of logarithms that \( \ln(e^{y}) = y \), we can simplify the left side:

\[
x = \ln(14)
\]

Thus, the exact solution is:

\[
x = \ln(14)
\]

Now, let's check the solution by substituting \( x = \ln(14) \) back into the original equation:

Original equation:
\[
e^{x} - 2 = 12
\]
Substituting \( x = \ln(14) \):

\[
e^{\ln(14)} - 2 = 12
\]

Since \( e^{\ln(14)} = 14 \), we have:

\[
14 - 2 = 12
\]
\[
12 = 12
\]

The left side equals the right side, confirming that our solution is correct.

Thus, the exact solution to the equation \( e^{x} - 2 = 12 \) is:

\[
\boxed{\ln(14)}
\]