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Chapter 3: Problem 51

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$2-6 \ln x=10$$

Short Answer

Expert verified
The solution to the equation is \(x = 0.262\)

Step by step solution

01

Isolate the logarithm

In order to start solving for \(x\), first isolate \(\ln x\) on one side of the equation by subtracting 2 from both sides of the equation. This gives us: \(-6 \ln x = 10 - 2\)\(-6 \ln x = 8\)
02

Remove the coefficient of the logarithm

Next, to completely isolate \(\ln x\), divide both sides of the equation by -6: \(\ln x = \frac{8}{-6}\)\(\ln x = -1.3333\)
03

Change from logarithmic to exponential form

To eliminate the logarithm, convert the equation from a logarithmic form to an exponential form. In general, the logarithmic equation \(\ln a = b\) can be rewritten in exponential form as, \(e^b = a\). By applying this: \(e^{-1.3333} = x\)
04

Approximate the result to three decimal places

Now use the approximation, to get the value of \(x\) to three decimal places: \(x = 0.262\)

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