Question
Solve the logarithmic equation. \( x=\log _{4} \sqrt[5]{4} \)
Ask by Cruz Munoz. in the United States
Feb 28,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = \frac{1}{5} \)
Solution
We start with the equation:
x = log₄(√[5]{4}).
Notice that the fifth root of 4 can be written in exponential form:
√[5]{4} = 4^(1/5).
Now, substitute into the logarithmic equation:
x = log₄(4^(1/5)).
Using the logarithmic identity log_b(a^c) = c · log_b(a), we get:
x = (1/5) · log₄(4).
Since log₄(4) equals 1 (because 4^1 = 4), this simplifies to:
x = 1/5.
Thus, the solution to the equation is:
x = 1/5.
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Simplify this solution Mind Expander
To solve the equation \( x = \log_{4} \sqrt[5]{4} \), we can rewrite \(\sqrt[5]{4}\) in terms of powers of 4: \(\sqrt[5]{4} = 4^{1/5}\). Now, substituting this back into the equation gives us: \[ x = \log_{4} (4^{1/5}) \] Using the power rule of logarithms, which states that \(\log_{b}(a^n) = n \cdot \log_{b}(a)\), we can simplify: \[ x = \frac{1}{5} \cdot \log_{4}(4) \] Since \(\log_{4}(4) = 1\): \[ x = \frac{1}{5} \cdot 1 = \frac{1}{5} \] Thus, the solution to the equation is: \( x = \frac{1}{5} \).
