$$ 1 \leftarrow \quad \begin{array}{l} ext { Solve the following logarithmic equation. Be sure to reject any value of } x ext { that is not in the domain of the } \ ext { original logarithmic expression. Give the exact answer. Then use a calculator to oblain a } \ ext { decimal approximation, correct to two decimal places, for the solution. } \ 6+8 \ln x=5\end{array} $$ Rewrite the given equation without logarithms. $$ \begin{array}{l} ext { Solve the equation What is the exact solution? Select the correct choice below and, if necessary, inl in the } \ ext { answer box to complete your choice. } \ ext { A. The solution set is }\{ ext { ? } \ ext { (Type an exact answer. Type your answer using exponential notation.) } \ ext { B. There are infinitely many solutions. }\end{array} $$

Question

$$ 1 \leftarrow \quad \begin{array}{l}	ext { Solve the following logarithmic equation. Be sure to reject any value of } x 	ext { that is not in the domain of the } \ 	ext { original logarithmic expression. Give the exact answer. Then use a calculator to oblain a } \ 	ext { decimal approximation, correct to two decimal places, for the solution. } \ 6+8 \ln x=5\end{array} $$ Rewrite the given equation without logarithms. $$ \begin{array}{l}	ext { Solve the equation What is the exact solution? Select the correct choice below and, if necessary, inl in the } \ 	ext { answer box to complete your choice. } \ 	ext { A. The solution set is }\{	ext { ? } \ 	ext { (Type an exact answer. Type your answer using exponential notation.) } \ 	ext { B. There are infinitely many solutions. }\end{array} $$

UpStudy AI · Accepted Answer

We begin with the equation
$$
6 + 8 \ln x = 5.
$$

**Step 1.** Isolate the logarithm term by subtracting 6 from both sides:
$$
8 \ln x = 5 - 6 = -1.
$$

**Step 2.** Divide both sides by 8:
$$
\ln x = -\frac{1}{8}.
$$

**Step 3.** Rewrite the equation without logarithms by exponentiating both sides (using the definition of the natural logarithm):
$$
x = e^{-\frac{1}{8}}.
$$

Since the domain of $$\ln x$$ requires $$x > 0$$, the solution $$x = e^{-\frac{1}{8}}$$ is valid.

**Exact Solution:**
$$
x = e^{-\frac{1}{8}}.
$$

**Decimal Approximation:**

Using a calculator,
$$
x \approx 0.88.
$$

Thus, the exact solution is $$x = e^{-\frac{1}{8}}$$, and the solution set is
$$
\{e^{-\frac{1}{8}}\}.
$$
The exact solution is $$ x = e^{-\frac{1}{8}} $$, and the solution set is $$ \{e^{-\frac{1}{8}}\} $$.

Rewrite the given equation without logarithms.

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