To solve the logarithmic equation [tex]\( \log(9x + 8) = 2 \)[/tex], we need to follow these steps:
1. Rewrite the logarithmic equation in its exponential form:
The logarithmic equation [tex]\(\log(9x + 8) = 2\)[/tex] can be rewritten as:
[tex]\[
10^2 = 9x + 8
\][/tex]
This is because the logarithm base 10 of a number is the exponent to which 10 must be raised to get that number.
2. Calculate the value of [tex]\(10^2\)[/tex]:
[tex]\[
10^2 = 100
\][/tex]
3. Set up the equation:
Now replace [tex]\(10^2\)[/tex] with 100 in the original equation:
[tex]\[
100 = 9x + 8
\][/tex]
4. Isolate the term containing [tex]\(x\)[/tex]:
To solve for [tex]\(x\)[/tex], we first need to isolate [tex]\(9x\)[/tex]. Subtract 8 from both sides of the equation:
[tex]\[
100 - 8 = 9x
\][/tex]
[tex]\[
92 = 9x
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
To find the value of [tex]\(x\)[/tex], divide both sides of the equation by 9:
[tex]\[
x = \frac{92}{9}
\][/tex]
6. Simplify the fraction (if possible):
The fraction [tex]\(\frac{92}{9}\)[/tex] can be simplified to its decimal form:
[tex]\[
x \approx 10.222222222222221
\][/tex]
Thus, the solution to the equation [tex]\(\log(9x + 8) = 2\)[/tex] is:
[tex]\[
x \approx 10.222222222222221
\][/tex]
