To solve the equation \(\log _4(x+20) = 3\), we can follow these steps:
1. Understand the problem: We need to solve for \(x\) where the logarithm base 4 of \(x + 20\) equals 3.
2. Convert the logarithmic equation to an exponential form: The logarithmic equation \(\log_b(a) = c\) is equivalent to the exponential equation \(b^c = a\). In this case:
[tex]\[
\log _4(x+20) = 3
\][/tex]
becomes:
[tex]\[
4^3 = x + 20
\][/tex]
3. Calculate the power: Compute \(4^3\):
[tex]\[
4^3 = 64
\][/tex]
4. Set up the equation: Now the equation is:
[tex]\[
64 = x + 20
\][/tex]
5. Solve for \(x\): Isolate \(x\) by subtracting 20 from both sides:
[tex]\[
x = 64 - 20
\][/tex]
[tex]\[
x = 44
\][/tex]
So the solution to the equation \(\log _4(x+20) = 3\) is:
[tex]\[
x = 44
\][/tex]
