Questions: Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then, use a calculator to obtain a decimal approximation for the solution. 3^(x+1)=3619 The solution set expressed in terms of logarithms is . 3. (Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the expression. Use ln for natural logarithm and log for common logarithm.)

Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then, use a calculator to obtain a decimal approximation for the solution. 3^(x+1)=3619 The solution set expressed in terms of logarithms is . 3. (Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the expression. Use ln for natural logarithm and log for common logarithm.)
Transcript text: Solve the following exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then, use a calculator to obtain a decimal approximation for the solution. \[ 3^{x+1}=3619 \] The solution set expressed in terms of logarithms is $\square$ 3. (Use a comma to separate answers as needed. Simplify your answer. Use integers or fractions for any numbers in the expression. Use In for natural logarithm and log for common logarithm.)
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Solution

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Solution Steps

Step 1: Take the Logarithm of Both Sides

To solve the equation \( 3^{x+1} = 3619 \), we take the logarithm of both sides. Using the common logarithm (base 10), we have:

\[ \log(3^{x+1}) = \log(3619) \]

Step 2: Apply the Power Rule of Logarithms

Using the power rule of logarithms, we can bring down the exponent:

\[ (x+1) \cdot \log(3) = \log(3619) \]

Step 3: Isolate \( x \)

Next, we isolate \( x \):

\[ x + 1 = \frac{\log(3619)}{\log(3)} \]

Subtracting 1 from both sides gives:

\[ x = \frac{\log(3619)}{\log(3)} - 1 \]

Step 4: Calculate the Decimal Approximation

Calculating the value of \( x \):

\[ x \approx 6.4585 \]

Final Answer

The solution set expressed in terms of logarithms is:

\[ \boxed{x = \frac{\log(3619)}{\log(3)} - 1} \]

The decimal approximation of the solution is:

\[ \boxed{x \approx 6.4585} \]

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