Pregunta
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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\( x = \frac{\ln(12)}{5 \ln(7)} \)
Solución
To solve the equation \( 5 \cdot 7^{5x} = 60 \) for \( x \), follow these steps:
1. **Isolate the Exponential Term:**
Start by dividing both sides of the equation by 5 to isolate the exponential expression:
\[
7^{5x} = \frac{60}{5} = 12
\]
2. **Apply the Natural Logarithm:**
Take the natural logarithm (\( \ln \)) of both sides to bring down the exponent:
\[
\ln(7^{5x}) = \ln(12)
\]
3. **Use Logarithm Power Rule:**
The power rule of logarithms states that \( \ln(a^b) = b \cdot \ln(a) \). Apply this to the left side:
\[
5x \cdot \ln(7) = \ln(12)
\]
4. **Solve for \( x \):**
Divide both sides by \( 5 \ln(7) \) to isolate \( x \):
\[
x = \frac{\ln(12)}{5 \ln(7)}
\]
5. **(Optional) Numerical Approximation:**
If you need a numerical value for \( x \), you can use a calculator to compute the natural logarithms:
\[
x \approx \frac{\ln(12)}{5 \ln(7)} \approx \frac{2.4849}{5 \times 1.9459} \approx \frac{2.4849}{9.7295} \approx 0.255
\]
So, \( x \approx 0.255 \).
**Final Answer:**
\[
x = \frac{\ln(12)}{5\, \ln(7)}
\]
Respondido por UpStudy AI y revisado por un tutor profesional
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Simplifique esta solución Beyond the Answer
To solve the equation \( 5(7)^{5x}=60 \), first, you can isolate the term with the exponent. Divide both sides by 5: \[ (7)^{5x} = \frac{60}{5} = 12 \] Next, you can take the natural logarithm of both sides, which allows you to bring down the exponent: \[ \ln((7)^{5x}) = \ln(12) \] This simplifies to: \[ 5x \cdot \ln(7) = \ln(12) \] Now, solve for \( x \): \[ x = \frac{\ln(12)}{5 \cdot \ln(7)} \] Substituting the approximate values of the logs: \[ x \approx \frac{2.4849}{5 \cdot 1.9459} \approx \frac{2.4849}{9.7295} \approx 0.255 \] So, the solution to the equation is \( x \approx 0.255 \).
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