To solve the exponential equation [tex]\(10^{5x + 8} = 9\)[/tex], we can follow these steps:
1. Take the logarithm of both sides of the equation.
[tex]\[
\log(10^{5x + 8}) = \log(9)
\][/tex]
2. Apply the properties of logarithms.
[tex]\[
(5x + 8) \cdot \log(10) = \log(9)
\][/tex]
3. Since [tex]\(\log(10) = 1\)[/tex], we simplify the equation:
[tex]\[
5x + 8 = \log(9)
\][/tex]
4. Isolate [tex]\(5x\)[/tex] by subtracting 8 from both sides:
[tex]\[
5x = \log(9) - 8
\][/tex]
5. Solve for [tex]\(x\)[/tex] by dividing both sides by 5:
[tex]\[
x = \frac{\log(9) - 8}{5}
\][/tex]
This represents the exact solution:
[tex]\[
x = \frac{\log(9) - 8}{5}
\][/tex]
Now, let's calculate the approximate value:
6. Compute the numerical value of [tex]\(\log(9)\)[/tex].
[tex]\[
\log(9) \approx 0.9542425094393249
\][/tex]
7. Substitute this value back into the equation for [tex]\( x \)[/tex]:
[tex]\[
x \approx \frac{0.9542425094393249 - 8}{5}
\][/tex]
8. Simplify the expression:
[tex]\[
x \approx \frac{-7.0457574905606751}{5}
\][/tex]
[tex]\[
x \approx -1.409151498112135
\][/tex]
9. Round the approximate value to 4 decimal places:
[tex]\[
x \approx -1.4092
\][/tex]
Summary of the solutions:
1. Exact solution:
[tex]\[
x = \frac{\log(9) - 8}{5}
\][/tex]
2. Approximation to 4 decimal places:
[tex]\[
x \approx -1.4092
\][/tex]
