Chapter 4: Problem 113

Graph the function. a. Graph \(Y_{1}=\log |x|\) and \(Y_{2}=\frac{1}{2} \log x^{2}\). How are the graphs related? b. Show algebraically that \(\frac{1}{2} \log x^{2}=\log |x|\).

### Short Answer

Expert verified

The graphs of \( Y_{1} = \log |x| \) and \( Y_{2} = \frac{1}{2} \log x^{2} \) are identical because \( Y_{2} \) simplifies to \( \log |x| \).

## Step by step solution

01

## - Understand the Functions

Identify and understand the given functions. The functions are given as:1. \( Y_{1} = \log |x| \) 2. \( Y_{2} = \frac{1}{2} \log x^{2} \).

02

## - Graph \( Y_{1} = \log |x| \)

Plot the graph of \( Y_{1} = \log |x| \). For positive values of \(x\), it behaves like \( \log x\), and for negative values of \(x\), it is the reflection of \(\log x\) about the y-axis. So, it is symmetrical about the y-axis.

03

## - Simplify \( Y_{2} = \frac{1}{2} \log x^{2} \)

Simplify the expression for \( Y_{2} \). Using the exponent property of logarithms: \( \log x^{2} = 2 \log |x| \), we can rewrite \( Y_{2} \) as \( Y_{2} = \frac{1}{2} (2 \log |x|) = \log |x| \).

04

## - Graph \( Y_{2} = \frac{1}{2} \log x^{2} \)

Since \( Y_{2} = \log |x| \), its graph will be the same as the graph of \( Y_{1} = \log |x| \). They are identical.

05

## - Relationship Between the Graphs

Since both functions simplify to \( \log |x| \), their graphs are exactly the same. This means that the graphs of \( Y_{1} \) and \( Y_{2} \) are identical.

## Key Concepts

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

###### Graphing logarithmic functions

Graphing logarithmic functions can sometimes be tricky, but it's crucial for understanding their behavior.

To graph the function \(Y_{1} = \log |x|\): For positive values of \(x\), it behaves just like \(\log x\). For negative values of \(x\), it mirrors the graph of \(\log x\) across the y-axis. This means the graph is symmetrical about the y-axis, which makes it easier to draw once one side is plotted.

You can now see the function's behavior across all x-values.

Next, we plot \(Y_{2} = \frac{1}{2} \log x^{2}\). Remember that \(x^{2}\) is always positive, so \(\log x^{2}\) is defined for all real numbers except zero.

As we simplify \(\frac{1}{2} \log x^{2}\), you'll find the graph of \(Y_{2}\) looks identical to \(\log |x|\).

Graphing these functions helps you see their relationships and understand how logarithms behave with various inputs.

###### Properties of logarithms

Understanding the properties of logarithms is vital for simplifying and manipulating these functions.

Here are some important properties:

**Product Property:**\(\log_b(xy) = \log_b(x) + \log_b(y)\)**Quotient Property:**\(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\)**Power Property:**\(\log_b(x^k) = k \log_b(x)\)

These properties make it straightforward to handle more complex logarithmic expressions.

In the exercise, we use the power property to simplify \(\log x^{2}\) into \(2 \log |x|\).

By applying the property correctly, it helps to graph and understand that both \(Y_{1} = \log |x|\) and \(Y_{2} = \frac{1}{2} \log x^{2}\) are actually the same function.

###### Absolute value in logarithms

When dealing with logarithmic functions, it's important to understand how absolute value changes things.

The absolute value function, \(|x|\), affects logarithmic functions by ensuring the input to the log is always positive.

For example, \(\log |x|\) is defined for all real numbers \(x eq 0\):

- If \(x\) is positive, \(\log |x|\) behaves like \(\log x\).
- If \(x\) is negative, \(\log |x|\) behaves like \(\log(-x)\).

This guarantees the input to the log remains valid.

In our exercise, both \(Y_{1}\) and \(Y_{2}\) incorporate the absolute value, making their domains broader and ensuring the functions are well-defined across all x-values (except zero).

Grasping the concept of absolute values in logarithms helps to plot these functions correctly and understand their symmetry.

###### Logarithmic identities

Logarithmic identities are powerful tools that simplify complex logarithmic expressions.

Some key logarithmic identities include:

**Change of Base Formula:**\(\log_b(x) = \frac{\log_k(x)}{\log_k(b)}\)**Identity Property:**\(\log_b(1) = 0\)**Inverse Property:**\(\log_b(b) = 1\)

These identities are especially useful in solving logarithmic equations or changing the base of a logarithm.

In our exercise, we used the property that \(\log x^{2}\) can be simplified to \(2 \log |x|\) to show that \(\frac{1}{2} \log x^{2}\) simplifies to \(\log |x|\).

Understanding and using logarithmic identities correctly helps in efficiently dealing with more complex logarithmic problems and spotting relationships between different functions.

This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept