Problem 146 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

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 1: 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 \(x^{2}\).

Step 2: 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 \((\ln x)^{2} = 2\ln x\).

Step 3: 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 \((\ln 1)^{2} = 2\ln 1 = 0\)

Step 4: 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.