To solve the equation [tex]\(8 = \log_b(256)\)[/tex] for [tex]\(b\)[/tex], follow these steps:
1. Rewrite the logarithmic equation in exponential form:
The equation [tex]\(8 = \log_b(256)\)[/tex] tells us that [tex]\(b\)[/tex] raised to the power of 8 equals 256. In other words,
[tex]\[
b^8 = 256.
\][/tex]
2. Solve for [tex]\(b\)[/tex] by taking the 8th root of both sides:
To find [tex]\(b\)[/tex], we need to isolate [tex]\(b\)[/tex]. We can do this by taking the 8th root of 256:
[tex]\[
b = \sqrt[8]{256}.
\][/tex]
3. Simplify the calculation:
Notice that 256 can be expressed as a power of 2. Specifically:
[tex]\[
256 = 2^8.
\][/tex]
Therefore,
[tex]\[
\sqrt[8]{256} = \sqrt[8]{2^8} = 2.
\][/tex]
So, the value of [tex]\(b\)[/tex] is:
[tex]\[
b = 2.
\][/tex]
Thus, the solution to the equation [tex]\(8 = \log_b(256)\)[/tex] is
[tex]\[
b = 2.
\][/tex]
