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Chapter 3: Problem 47

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$2.1=\ln 6 x$$

Short Answer

Expert verified
The solution to the equation \(2.1 = \ln 6x\) is approximately \(x = 0.753\).

Step by step solution

01

Isolate the logarithm

Subtract 0 from both sides of the equation to isolate the \(\ln 6x\) on one side: \[2.1 = \ln 6x\] This operation will result in the same initial equation.
02

Convert the logarithmic equation to its equivalent exponential form

Using the property of logarithms that if \(a = \ln b\), then \(b = e^a\), you can rewrite the equation as \[6x = e^{2.1}\]
03

Solve for \(x\)

Now, divide both sides of the equation by 6 to solve for \(x\): \[x = \frac{e^{2.1}}{6}\]
04

Simplify and round to three decimal places

Using a calculator, calculate \(e^{2.1}\) divided by 6 and round to three decimal places: \[x \approx 0.753\]

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Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$\ln (x+1)=2-\ln x$$

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