Questions: When solving the equation 3^x=5 choose all correct answers Take the natural logarithm of both sides to get ln(3^x)=ln(5) which simplifies to x=ln(5) Use the base 10 logarithm and the property log(m^n)=n log(m) to get log(3^x)=log(5) which simplifies to x log(3)=log(5) Then divide both sides by log(3) Use the natural logarithm and the property ln(m^n)=n ln(m) to get ln(3^x)=ln(5) which simplifies to x ln(3)=ln(5) Then divide both sides by ln(3) Raise both sides to the 1/x power to get 3=5^(1/x)

When solving the equation 3^x=5 choose all correct answers Take the natural logarithm of both sides to get ln(3^x)=ln(5) which simplifies to x=ln(5) Use the base 10 logarithm and the property log(m^n)=n log(m) to get log(3^x)=log(5) which simplifies to x log(3)=log(5) Then divide both sides by log(3) Use the natural logarithm and the property ln(m^n)=n ln(m) to get ln(3^x)=ln(5) which simplifies to x ln(3)=ln(5) Then divide both sides by ln(3) Raise both sides to the 1/x power to get 3=5^(1/x)
Transcript text: When solving the equation $3^{x}=5$ choose all correct answers Take the natural logarithm of both sides to get $\ln \left(3^{x}\right)=\ln (5)$ which simplifies to $x=\ln (5)$ Use the base 10 logarithm and the property $\log \left(m^{n}\right)=n \log (m)$ to get $\log \left(3^{x}\right)=\log (5)$ which simplifies to $x \log (3)=\log (5)$ Then divide both sides by $\log (3)$ Use the natural logarithm and the property $\ln \left(m^{n}\right)=n \ln (m)$ to get $\ln \left(3^{x}\right)=\ln (5)$ which simplifies to $x \ln (3)=\ln (5)$ Then divide both sides by $\ln (3)$ Raise both sides to the $\frac{1}{x}$ power to get \[ 3=5^{1 / x} \]
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Solution

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Solution Steps

To solve the equation \(3^x = 5\), we can use logarithms to isolate \(x\). Here are the steps:

  1. Take the natural logarithm (ln) of both sides of the equation.
  2. Use the property of logarithms \(\ln(a^b) = b \ln(a)\) to simplify.
  3. Solve for \(x\) by isolating it on one side of the equation.
Step 1: Take the Natural Logarithm of Both Sides

Given the equation \(3^x = 5\), we take the natural logarithm of both sides: \[ \ln(3^x) = \ln(5) \]

Step 2: Apply the Logarithm Power Rule

Using the property of logarithms \(\ln(a^b) = b \ln(a)\), we can simplify the left-hand side: \[ x \ln(3) = \ln(5) \]

Step 3: Solve for \(x\)

To isolate \(x\), we divide both sides by \(\ln(3)\): \[ x = \frac{\ln(5)}{\ln(3)} \]

Step 4: Calculate the Value of \(x\)

Using the values of the natural logarithms: \[ \ln(5) \approx 1.6094 \quad \text{and} \quad \ln(3) \approx 1.0986 \] we get: \[ x \approx \frac{1.6094}{1.0986} \approx 1.465 \]

Final Answer

\[ \boxed{x = \frac{\ln(5)}{\ln(3)}} \]

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