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Chapter 5: Problem 67

Simplify. $$\log _{e} e^{|x-4|}$$

Short Answer

Expert verified
\( |x-4| \)

Step by step solution

01

Understand the problem

The problem requires simplifying the logarithmic expression \( \log _{e} e^{|x-4|} \). The base of the logarithm is e, which is the natural logarithm (ln).
02

Apply the property of logarithms

Recall the property of logarithms: \( \log _{b} b^{y} = y \). Since the base of the logarithm and the base of the exponent are the same (e), we can use this property directly.
03

Simplify the expression

Using the property mentioned, the expression simplifies to \( |x-4| \). So, \( \log _{e} e^{|x-4|} = |x-4| \).

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Key Concepts

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

logarithmic properties
When solving logarithmic problems, it's essential to understand some important properties of logarithms. These properties help in simplifying complex logarithmic expressions.
A crucial property to know is that for any base, \(\b\) and any number, \(\):
  • \(\log_{b} b^n = n\)
This means the logarithm of a base raised to an exponent simplifies to the exponent itself.
For example, \(\log_{e} e^5 = 5\).
In our problem, we used this property to simplify \(\log _{e} e^{|x-4|}\) to \(|x-4|\), since the base of the logarithm (e) and the base of the exponent (e) are the same.
absolute value
Understanding absolute value is fundamental in handling the given expression. The absolute value function, denoted by \(|x|\), represents the distance of a number from zero on the number line irrespective of direction.
Therefore, \(|x-4|\) represents how far the value of \(x\) is from 4 without considering which side of 4 it lies on.
For example, if \(x = 6\), then \(|x-4| = 2\), and if \(x = 2\), still \(|x-4| = 2\).
The absolute value ensures that the result is always non-negative.
simplification
Simplification in this context involves expressing the logarithmic function in a more straightforward form. Here's how we approached it:
  • Identify the bases - We noticed the base of the logarithm and the exponent were the same (e).
  • Apply the logarithmic property - We used the property \(\log_{b} b^y = y\) to simplify the expression directly.
  • Result - Simplifying \(\log _{e} e^{|x-4|}\) using the property mentioned, resulted in \(|x-4|\).
By following these steps, we transformed the complex expression into a simpler form that is much easier to understand and work with.

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