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Chapter 4: Problem 146

Check each proposed solution by direct substitution or with a graphing utility. $$(\ln x)^{2}=\ln x^{2}$$

Short Answer

Expert verified
Direct substitutions proved to show that the original equality holds. The verification can be further supported by comparing the graphs of the two functions.

Step by step solution

01

Understand the formulas

To get started, it's important to understand what \((\ln x)^{2}\) and \(\ln x^{2}\) mean. \((\ln x)^{2}\) means the natural logarithm of \(x\), squared. On the other hand, \(\ln x^{2}\) means the natural logarithm of \(x^{2}\).
02

Simplify \(\ln x^{2}\)

Take advantage of the logarithmic property \( \ln a^{b} = b \ln a\) to simplify \(\ln x^{2}\) into \(2\ln x\). So, the equation becomes \((\ln x)^{2} = 2\ln x\).
03

Direct Substitution

Let's use direct substitution with some values of \(x\) to check if the two sides of the equation are equal. For instance, let \(x = 1\). Then both sides of the equation become \((\ln 1)^{2} = 2\ln 1 = 0\)
04

Graphing Functions

Plotting the two functions will can provide a visual comparison. Using a graphing utility, plot \(y = (\ln x)^{2}\) and \(y = 2\ln x\). If the two graphs coincide, then the original equation holds true.

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