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Chapter 8: Problem 6

Solve (a) \(\ln \left(2 x^{2}\right)=3\) (b) \(2 \ln \left(x^{2}\right)=3\) (c) \(\ln \left(x^{2}+2\right)=1.3\) (d) \(3 \ln \left(x^{2}+1\right)=3.9\)

Short Answer

Expert verified
Question: Solve the following logarithmic equations: (a) \(\ln \left(2 x^{2}\right)=3\) (b) \(2 \ln \left(x^{2}\right)=3\) (c) \(\ln \left(x^{2}+2\right)=1.3\) (d) \(3 \ln \left(x^{2}+1\right)=3.9\) Answer: (a) \(x = \pm \sqrt{\frac{e^3}{2}}\) (b) \(x = \pm \sqrt{e^{\frac{3}{2}}}\) (c) \(x = \pm \sqrt{e^{1.3} - 2}\) (d) \(x = \pm \sqrt{e^{\frac{3.9}{3}} - 1}\)

Step by step solution

01

Eliminate the natural logarithm

Use the exponential function to eliminate the natural logarithm to get: \(2 x^2 = e^3\)
02

Solve for x

Divide both sides by 2 to obtain: \(x^2 = \frac{e^3}{2}\). Now take the square root of both sides, considering both positive and negative solutions: \(x = \pm \sqrt{\frac{e^3}{2}}\) (b) \(2 \ln \left(x^{2}\right)=3\)
03

Eliminate the natural logarithm

Divide both sides by 2 to get: \(\ln \left(x^{2}\right) = \frac{3}{2}\). Use the exponential function to eliminate the natural logarithm to get: \(x^2 = e^{\frac{3}{2}}\)
04

Solve for x

Take the square root of both sides, considering both positive and negative solutions: \(x = \pm \sqrt{e^{\frac{3}{2}}}\) (c) \(\ln \left(x^{2}+2\right)=1.3\)
05

Eliminate the natural logarithm

Use the exponential function to eliminate the natural logarithm to get: \(x^2 + 2 = e^{1.3}\)
06

Solve for x

Subtract 2 from both sides: \(x^2 = e^{1.3} - 2\). Now take the square root of both sides, considering both positive and negative solutions: \(x = \pm \sqrt{e^{1.3} - 2}\) (d) \(3 \ln \left(x^{2}+1\right)=3.9\)
07

Eliminate the natural logarithm

Divide both sides by 3 to get: \(\ln \left(x^{2}+1\right) = \frac{3.9}{3}\). Use the exponential function to eliminate the natural logarithm to get: \(x^2 + 1 = e^{\frac{3.9}{3}}\)
08

Solve for x

Subtract 1 from both sides: \(x^2 = e^{\frac{3.9}{3}} - 1\). Now take the square root of both sides, considering both positive and negative solutions: \(x = \pm \sqrt{e^{\frac{3.9}{3}} - 1}\)

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Most popular questions from this chapter

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