Question

Solve the exponential equation. Round your answer to the nearest thousandth.




8
(
5
x
)
−
14
=
78
8(5
x
)−14=78




x
=
x=

Answer 1

To solve the equation \( 8(5^x) - 14 = 78 \), we first isolate the term involving the exponent:

1. Add 14 to both sides:
\[
8(5^x) = 78 + 14
\]
\[
8(5^x) = 92
\]

2. Divide both sides by 8:
\[
5^x = \frac{92}{8}
\]
\[
5^x = 11.5
\]

3. Now, we take the natural logarithm of both sides:
\[
\ln(5^x) = \ln(11.5)
\]

4. Utilize the property of logarithms that allows us to move the exponent in front:
\[
x \cdot \ln(5) = \ln(11.5)
\]

5. Solve for \( x \):
\[
x = \frac{\ln(11.5)}{\ln(5)}
\]

Now we can calculate \( x \):

Using a calculator, we find:
- \( \ln(11.5) \approx 2.4423 \)
- \( \ln(5) \approx 1.6094 \)

Thus,
\[
x \approx \frac{2.4423}{1.6094} \approx 1.519
\]

Rounding to the nearest thousandth, we conclude:
\[
\boxed{1.519}
\]