To solve the equation [tex]\(\log_2(3x + 8) = 5\)[/tex], let's break down the problem step-by-step.
1. Understand the given logarithmic equation:
The equation [tex]\(\log_2(3x + 8) = 5\)[/tex] is in logarithmic form where the base is 2.
2. Convert the logarithmic equation to its exponential form:
The general property of logarithms states that if [tex]\(\log_b(a) = c\)[/tex], then [tex]\(b^c = a\)[/tex]. Applying this property here:
[tex]\[
\log_2(3x + 8) = 5 \implies 2^5 = 3x + 8
\][/tex]
3. Simplify the exponential equation:
Calculate [tex]\(2^5\)[/tex]:
[tex]\[
2^5 = 32
\][/tex]
So, the equation becomes:
[tex]\[
32 = 3x + 8
\][/tex]
4. Isolate the variable [tex]\(x\)[/tex]:
To solve for [tex]\(x\)[/tex], we need to isolate [tex]\(x\)[/tex] on one side of the equation. First, subtract 8 from both sides of the equation:
[tex]\[
32 - 8 = 3x
\][/tex]
Simplifying this:
[tex]\[
24 = 3x
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Now, divide both sides by 3 to get [tex]\(x\)[/tex]:
[tex]\[
x = \frac{24}{3} = 8
\][/tex]
6. Write the complete solution:
The solution to the equation [tex]\(\log_2(3x + 8) = 5\)[/tex] is:
[tex]\[
x = 8
\][/tex]
Therefore, the final answer is:
[tex]\[
\boxed{8}
\][/tex]
