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\( m = \frac{\log_{3}{2}}{2} \)
Solución
Solve the equation \( 3^{4m}=4 \).
Solve the equation by following steps:
- step0: Solve for \(m\):
\(3^{4m}=4\)
- step1: Take the logarithm of both sides:
\(\log_{3}{\left(3^{4m}\right)}=\log_{3}{\left(4\right)}\)
- step2: Evaluate the logarithm:
\(4m=\log_{3}{\left(4\right)}\)
- step3: Divide both sides:
\(\frac{4m}{4}=\frac{\log_{3}{\left(4\right)}}{4}\)
- step4: Divide the numbers:
\(m=\frac{\log_{3}{\left(4\right)}}{4}\)
- step5: Simplify:
\(m=\frac{\log_{3}{\left(2\right)}}{2}\)
The solution to the equation \(3^{4m}=4\) is \(m=\frac{\log_{3}{(2)}}{2}\).
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Simplifique esta solución Beyond the Answer
To solve for \( m \) in the equation \( 3^{4m} = 4 \), start by taking the logarithm of both sides: \[ \log(3^{4m}) = \log(4) \] Using the power rule of logarithms, this simplifies to: \[ 4m \cdot \log(3) = \log(4) \] Now, isolate \( m \): \[ m = \frac{\log(4)}{4 \cdot \log(3)} \] Now, you can calculate \( m \) using a calculator. Using common logarithms (base 10), \[ \log(4) \approx 0.6021 \quad \text{and} \quad \log(3) \approx 0.4771 \] Substituting these approximations in: \[ m \approx \frac{0.6021}{4 \cdot 0.4771} \approx \frac{0.6021}{1.9084} \approx 0.315 \] Thus, \( m \approx 0.315 \).
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