Question

solve the following logarithmic equation, using a calculator if necessary to evaluate the logarithm. write your answer as a fraction or round your answer to two decimal places. (e ≈ 2.7183 )
ln(7x + 12) = 3
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Explanation:

Step1: Exponentiate both sides with base \( e \)

To solve the logarithmic equation \( \ln(7x + 12) = 3 \), we use the property that if \( \ln(a)=b \), then \( e^{\ln(a)} = e^{b} \), which simplifies to \( a = e^{b} \). Applying this to our equation, we get:
\( e^{\ln(7x + 12)} = e^{3} \)
Simplifying the left - hand side (since \( e^{\ln(u)}=u \) for \( u>0 \)), we have:
\( 7x + 12 = e^{3} \)

Step2: Solve for \( x \)

First, we know that \( e\approx2.7183 \), so \( e^{3}\approx2.7183^{3}\approx20.0855 \).
Now, we solve the equation \( 7x+12 = e^{3} \) for \( x \). Subtract 12 from both sides:
\( 7x=e^{3}-12 \)
Then divide both sides by 7:
\( x=\frac{e^{3}-12}{7} \)
Substitute \( e^{3}\approx20.0855 \) into the formula:
\( x=\frac{20.0855 - 12}{7}=\frac{8.0855}{7}\approx1.155 \approx 1.16 \) (rounded to two decimal places)

Answer:

\( x\approx1.16 \) (or if we use the exact fraction form \( x = \frac{e^{3}-12}{7}\), but the problem asks to round to two decimal places or write as a fraction. If we calculate the exact value: \( e^{3}=20.0855369232\), \( e^{3}-12 = 8.0855369232\), \( \frac{8.0855369232}{7}\approx1.155\approx1.16 \))