Question
Solve the following exponential equation. Exact solutions only.
1
2
e
(
x
−
5
)
=
13
2
1
e
(x−5)
=13
x
=
x=
1
2
e
(
x
−
5
)
=
13
2
1
e
(x−5)
=13
x
=
x=
Answers
GPT-4o mini
To solve the exponential equation \( \frac{1}{2} e^{(x-5)} = 13 \), we can follow these steps:
1. **Isolate the exponential term**:
\[
e^{(x-5)} = 13 \cdot 2
\]
This simplifies to:
\[
e^{(x-5)} = 26
\]
2. **Take the natural logarithm of both sides**:
\[
\ln(e^{(x-5)}) = \ln(26)
\]
Using the property of logarithms, \( \ln(e^y) = y \), we can simplify the left side:
\[
x - 5 = \ln(26)
\]
3. **Solve for \( x \)**:
\[
x = \ln(26) + 5
\]
Thus, the exact solution for \( x \) is:
\[
x = \ln(26) + 5
\]
1. **Isolate the exponential term**:
\[
e^{(x-5)} = 13 \cdot 2
\]
This simplifies to:
\[
e^{(x-5)} = 26
\]
2. **Take the natural logarithm of both sides**:
\[
\ln(e^{(x-5)}) = \ln(26)
\]
Using the property of logarithms, \( \ln(e^y) = y \), we can simplify the left side:
\[
x - 5 = \ln(26)
\]
3. **Solve for \( x \)**:
\[
x = \ln(26) + 5
\]
Thus, the exact solution for \( x \) is:
\[
x = \ln(26) + 5
\]