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Chapter 7: Problem 5

Using rules of logarithms, convert each equation to its power functionequivalent in the form \(y=k x^{p}\). a. \(\log y=\log 4+2 \log x\) c. \(\log y=\log 1.25+4 \log x\) b. \(\log y=\log 2+4 \log x\) d. \(\log y=\log 0.5+3 \log x\)

### Short Answer

Expert verified

a. \( y = 4 x^2 \), b. \( y = 1.25 x^4 \), c. \( y = 2 x^4 \), d. \( y = 0.5 x^3 \).

## Step by step solution

01

## Identify Logarithmic Properties

Recognize the logarithmic properties to be used. In these cases, the properties used are \(\text{logarithm of a product}\) and \(\text{logarithm of a power}\). The properties are: \(\text{log}(a \times b) = \text{log} a + \text{log} b\) and \(\text{log}(a^b) = b \times \text{log} a\).

02

## Expand the Logarithmic Equations

Rewrite the given equations using the properties identified. Specifically: \ a. \(\text{log} y = \text{log} 4 + 2 \text{log} x\) expands to \(\text{log} y = \text{log} (4 \times x^2)\) \ b. \(\text{log} y = \text{log} 1.25 + 4 \text{log} x\) expands to \(\text{log} y = \text{log} (1.25 \times x^4)\) \ c. \(\text{log} y = \text{log} 2 + 4 \text{log} x\) expands to \(\text{log} y = \text{log} (2 \times x^4)\) \ d. \(\text{log} y = \text{log} 0.5 + 3 \text{log} x\) expands to \(\text{log} y = \text{log} (0.5 \times x^3)\).

03

## Solve for y

Remove the logarithm on both sides of the equations using the property \( \text{log} a = \text{log} b \implies a = b \). This gives: \ a. \( y = 4 \times x^2 \) \ b. \( y = 1.25 \times x^4 \) \ c. \( y = 2 \times x^4 \) \ d. \( y = 0.5 \times x^3 \).

## Key Concepts

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

###### exponential functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions are powerful tools in describing growth or decay in fields such as biology, economics, and physics.

Each exponential function takes the form: \( y = k x^p \). In this form, \( k \) is a constant and \( p \) represents the power to which the variable \( x \) is raised.

For example, for the equation \( y = 4 x^2 \), 4 is the constant factor (\( k \)), and 2 is the exponent (\( p \)). This indicates that \( y \) grows quadratically as \( x \) changes.

Understanding how to convert logarithmic functions to exponential functions allows students to grasp the behavior and graph of functions efficiently.

- You see this conversion in everyday scenarios, such as calculating compound interest or population growth.
- The ability to switch between these forms simplifies solving for \( y \) and provides insights into the underlying relationship between variables.

###### logarithm of a product

The logarithm of a product property simplifies the multiplication of numbers within a logarithmic function. This property states: \( \log(a \times b) = \log a + \log b \).

Imagine you need to solve \( \log (4 \times x^2) \). Instead of directly computing the product, you can use the property to break it down:

- Identify and separate the components: \( \log y = \log 4 + \log (x^2) \)
- This separation makes dealing with more complex expressions easier.

Let's apply this to one of our given equations:

For \( \log y = \log 4 + 2 \log x \):

- Using the product property, this changes to \( \log y = \log (4 \times x ^2) \).

This step is crucial in converting the logarithmic form back into its exponential form. Recognizing and using the logarithm of a product property simplifies many problems and clarifies the relationships between different parts of an equation.

###### logarithm of a power

The logarithm of a power property is fundamental in simplifying expressions where a logarithm involves an exponent. This property says: \( \log(a^b) = b \times \log a \).

This means that you can take the exponent out in front as a multiplier. For example, in the expression \( \log(x^2) \):

- Using the power property, this can be rewritten as \( 2 \log x \).

Applying this to our original equations:

In \( \log y = \log 4 + 2 \log x \):

- We notice the \( 2 \log x \) term. Convert \( 2 \log x\) to \( \log(x^2) \): \( \log y = \log 4 + \log(x^2) \).
- Next, use the logarithm of a product property to combine terms: \( \log y = \log (4 \times x^2) \). This further simplifies to its exponential form: \( y = 4 \times x^2 \).

The logarithm of a power property streamlines complex logarithmic expressions. It is invaluable in turning them into simpler, more manageable forms. Recognizing when and how to use this property can significantly simplify solving logarithmic equations.

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