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Chapter 4: Problem 29

Solve the following equations for \(x.\) $$\ln (\ln 3 x)=0$$

Short Answer

Expert verified
The solution is \(x = \frac{e}{3}\).

Step by step solution

01

Understand the equation

The given equation is \(\text{ln} (\text{ln} 3x) = 0\). This equation involves a nested natural logarithm.
02

Isolate the inner logarithm

First, recognize that for \(\text{ln} y = 0\), we need \(y = 1\). Therefore, set the inner logarithm equal to 1: \(\text{ln} 3x = 1\).
03

Solve the inner equation

To solve \(\text{ln} 3x = 1\), exponentiate both sides (use the natural base \(e\)): \(3x = e^1 \). So, \(3x = e\).
04

Solve for \(x\)

Divide both sides of the equation \(3x = e\) by 3 to isolate \(x\): \(x = \frac{e}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

nested natural logarithm
Nested natural logarithms can seem complicated, but breaking them down step by step helps to simplify the process. The natural logarithm (ln) function is the inverse of the exponential function, which uses the base 'e' (approximately 2.718). Considering a nested logarithm means you have a logarithm inside another logarithm. In the equation \(\ln(\ln(3x)) = 0\), the outer logarithm is applied to the inner logarithm \(\ln(3x)\). To solve for 'x', you first need to simplify the inner part before dealing with the outer.
isolating the variable
Isolating the variable 'x' involves several steps where we gradually simplify the equation. Starting with the inner logarithm, from \(\ln(3x) = 1\), we use the property that \ln(y) = 0\ means \y = 1\. This step allows us to isolate \(\ln(3x)\), making our new equation \(\ln(3x) = 1\). This simplification step is crucial because it transitions our complicated nested logarithm to a more straightforward form.
exponentiation
Exponentiation is the process of raising a number to a certain power and is the reverse operation of taking a logarithm. With our isolated inner logarithm \(\ln(3x) = 1\), exponentiating both sides (raising 'e' to the power of both sides) eliminates the logarithm. This gives us \[3x = e\]. To find 'x', we simply divide by 3, resulting in \x = \frac{e}{3}\. This step converts the logarithmic form back to a straightforward algebraic equation that can be easily solved.

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