Question

solve the following logarithmic equation. be sure to reject any value of x that is not in the domain of the original logarithmic expression. give the exact answer. then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 6 + 8 ln x = 5 rewrite the given equation without logarithms.

Explanation:

Step1: Isolate the logarithmic term

Subtract 6 from both sides of the equation \(6 + 8\ln x = 5\) to get \(8\ln x = 5 - 6\).
\(8\ln x = -1\)

Step2: Solve for \(\ln x\)

Divide both sides by 8: \(\ln x=\frac{-1}{8}\)

Step3: Convert to exponential form

Recall that \(\ln x = y\) is equivalent to \(x = e^y\). So, \(x = e^{-\frac{1}{8}}\)

Step4: Approximate the decimal value

Calculate \(e^{-\frac{1}{8}}\approx e^{-0.125}\approx0.882\) (using a calculator, rounded to two decimal places)

Answer:

Exact answer: \(x = e^{-\frac{1}{8}}\)
Decimal approximation: \(x\approx0.88\)