To solve the equation [tex]\(\log_4(x + 20) = 3\)[/tex], we'll use the definition and properties of logarithms. Here is the step-by-step process:
1. Understand the logarithmic equation:
[tex]\[
\log_4(x + 20) = 3
\][/tex]
This equation states that the logarithm of [tex]\(x + 20\)[/tex] with base 4 is equal to 3.
2. Rewrite the logarithm in exponential form:
According to the definition of a logarithm, if [tex]\(\log_b(a) = c\)[/tex] then [tex]\(b^c = a\)[/tex]. Therefore:
[tex]\[
4^3 = x + 20
\][/tex]
3. Calculate the value of [tex]\(4^3\)[/tex]:
[tex]\[
4^3 = 4 \times 4 \times 4 = 64
\][/tex]
4. Set up the equation with the computed value:
[tex]\[
64 = x + 20
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Subtract 20 from both sides of the equation:
[tex]\[
x = 64 - 20
\][/tex]
6. Perform the subtraction:
[tex]\[
x = 44
\][/tex]
Therefore, the solution to the equation [tex]\(\log_4(x + 20) = 3\)[/tex] is:
[tex]\[
x = 44
\][/tex]
[tex]\(\boxed{44}\)[/tex]
