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Chapter 3: Problem 96

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$2+3 \ln x=12$$

Short Answer

Expert verified
The solution to the logarithmic equation given is \(x \approx 20.086\).

Step by step solution

01

Subtract 2 from both sides

Subtract 2 from both sides of the equation to isolate the logarithm, which gives the following equation: \(3 \ln x=12 - 2\)\(3 \ln x=10\)
02

Divide by 3

Divide through by 3 to get:\(\ln x = \frac{10}{3}\)
03

Exponentiate

Getting rid of the logarithm, \(x\), is achieved by exponentiating the equation by base \(e\). This is due to the rule that for any positive number \(b\), \(\ln b = x\) then the inverse operation removes the logarithm, i.e., \(e^{x}=b\). So the equation becomes: \(x = e^{10/3}\)
04

Approximate the result

Using a calculator to find \(e^{10/3}\) (here, \(e\) is the base of the natural logarithm, approximately equal to 2.71828), it is found that the value of \(x\) approximated to three decimal places is \(x\approx 20.086\).

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