Problem 148 - College Algebra Essentials

Question

Check each proposed solution by direct substitution or with a graphing utility. $$\ln (\ln x)=0$$

Step-by-Step Solution

Answer

The solution to the equation \(\ln (\ln x) = 0\), is \(x \approx 2.71828\) or \(x = e\), where \(e\) is Euler's number. This was confirmed by substituting \(x = e\) back into the original equation and confirming the equality holds.

Step 1: Understand the equation format

Given the logarithmic equation \(\ln (\ln x) = 0\). The challenge here is to convert this logarithmic form into an equivalent form that can facilitate solving for the value of \(x\). First, we need to understand the idea behind logarithmic form: the equation \(\ln a = b\) can be rewritten in exponential form as \(e^b = a\), where \(e\) denotes Euler's number, approximately equal to 2.71828. This is the base of the natural logarithm.

Step 2: Convert the equation from logarithmic to exponential form

Applying the logic mentioned in Step 1, we can rewrite the equation \(\ln (\ln x) = 0\) in exponential form. We achieve this by setting \(0\) as the exponent of \(e\), giving us \(e^0 = \ln x\). Simplifying \(e^0\) which equals \(1\), we get \(\ln x = 1\). Therefore, the equation simplifies to asking for which \(x\), the natural logarithm of \(x\) equals \(1\).

Step 3: Solve for \(x\)

Next, convert \(\ln x = 1\) back into exponential form to directly solve for \(x\). Applying the same change we did in step 2 gives us: \(e^1 = x\). Thus, \(x = e\). Given the base of the natural logarithm \(e\) is approximately 2.71828, we can conclude that \(x \approx 2.71828\). Check this by direct substitution into the original equation.

Step 4: Check the solution by substituting in the original equation

Our final step involves substituting \(x = e\) into the original logarithmic equation to check that the solution is correct. Substituting \(x = e\) into the equation \(\ln (\ln x) = 0\), we get \(\ln (\ln e) = 0\). \( \ln e = 1\) and thus we have \(\ln 1 = 0\). \(\ln 1 = 0\) is a correct equation. This shows that our solution, \(x = e\), is indeed correct.