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Solve. \( \log _{3}(v)=-4 \) \( v= \) Question Help: Dideo Submit Question
Ask by Todd Page. in the United States
Mar 18,2025
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Tutor-Verified Answer
Answer
\( v=\frac{1}{81} \)
Solution
Solve the equation \( \log_{3}(v)=-4 \).
Solve the equation by following steps:
- step0: Solve for \(v\):
\(\log_{3}{\left(v\right)}=-4\)
- step1: Find the domain:
\(\log_{3}{\left(v\right)}=-4,v>0\)
- step2: Convert the logarithm into exponential form:
\(v=3^{-4}\)
- step3: Evaluate the power:
\(v=\frac{1}{81}\)
- step4: Check if the solution is in the defined range:
\(v=\frac{1}{81},v>0\)
- step5: Find the intersection:
\(v=\frac{1}{81}\)
The solution to the equation \( \log_{3}(v)=-4 \) is \( v=\frac{1}{81} \).
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Simplify this solution The Deep Dive
To solve the equation \( \log_{3}(v) = -4 \), we need to rewrite this logarithmic equation in its exponential form. This means \( v \) is equal to the base (which is 3) raised to the power of the right side (-4). So, we can express it as: \( v = 3^{-4} \) Now, calculating \( 3^{-4} \): \( 3^{-4} = \frac{1}{3^4} = \frac{1}{81} \) Therefore, the solution is: \( v = \frac{1}{81} \)
