Moran Peterson
12/26/2023 · Senior High School
Solve: \( 4=\log _{b}(625) \)
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Tutor-Verified Answer
Step-by-step Solution
Solve the equation \( 4=\log_{b}(625) \).
Solve the equation by following steps:
- step0: Solve for \(b\):
\(4=\log_{b}{\left(625\right)}\)
- step1: Find the domain:
\(4=\log_{b}{\left(625\right)},b \in \left(0,1\right)\cup \left(1,+\infty\right)\)
- step2: Swap the sides:
\(\log_{b}{\left(625\right)}=4\)
- step3: Convert the logarithm into exponential form:
\(625=b^{4}\)
- step4: Swap the sides:
\(b^{4}=625\)
- step5: Simplify the expression:
\(b=\pm \sqrt[4]{625}\)
- step6: Simplify:
\(b=\pm 5\)
- step7: Separate into possible cases:
\(\begin{align}&b=5\\&b=-5\end{align}\)
- step8: Check if the solution is in the defined range:
\(\begin{align}&b=5\\&b=-5\end{align},b \in \left(0,1\right)\cup \left(1,+\infty\right)\)
- step9: Find the intersection:
\(b=5\)
The solution to the equation \(4=\log_{b}(625)\) is \(b=5\).
Quick Answer
\( b=5 \)
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