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Chapter 4: Problem 78

Write each equation as an equivalent logarithmic equation. $$w^{x+2}=r$$

Short Answer

Expert verified
\(\text{log}_w(r)=x+2\)

Step by step solution

01

Understand the Exponential Form

The given equation is in exponential form: y^{x+2}=r
02

Identify the Base, Exponent, and Result

In the equation, identify the base (w), the exponent (x+2), and the result (r).
03

Convert to Logarithmic Form

Write the equivalent logarithmic equation by isolating the exponent. The exponential form y^{x+2}=ryields the logarithmic form: \(\text{log}_w(r)=x+2\)
04

Simplify if Necessary

Since the exponent in the logarithmic equation is already simplified, the final equivalent logarithmic equation is: \(\text{log}_w(r) = x + 2\)

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Key Concepts

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

exponential form
Exponential form is a way to express repeated multiplication by the same factor. In an equation like the one given, \( w^{x+2} = r \), the term \( w^{x+2} \) is in exponential form. This means that the base \( w \) is raised to the power of the exponent \( x+2 \). Understanding this form is crucial because it helps us convert equations into other forms, such as logarithmic form.
base and exponent
In our equation \( w^{x+2} = r \), it鈥檚 important to identify the base and the exponent.
鈥� The base is the number or variable that is being multiplied by itself. In this case, the base is \( w \).
鈥� The exponent is the power to which the base is raised. Here, the exponent is \( x+2 \).
Knowing the base and exponent makes it easier to understand and manipulate the equation, especially when converting between forms like exponential and logarithmic.
logarithmic form conversion
Converting from exponential form to logarithmic form involves isolating the exponent. The given exponential equation \( w^{x+2} = r \) converts to the logarithmic form by expressing the exponent as a logarithm.
The general formula is: \[ b^y = x \ \text{becomes} \ \text{log}_b(x) = y \] when applied to our equation: \[ w^{x+2} = r \ \text{becomes} \ \text{log}_w(r) = x+2 \] This tells us what power (\( x+2 \)) the base (\( w \)) must be raised to in order to produce the result (\( r \)). Logarithmic form is useful for solving equations where the exponent is unknown.
simplification of equations
Once we have converted the exponential equation to a logarithmic equation, simplification is the next step. For the equation \( w^{x+2} = r \), the resulting logarithmic form is: \[ \text{log}_w(r) = x+2 \]
In this case, the equation is already simplified since the exponent is isolated. Often, simplification involves rearranging the equation to isolate the variable of interest.
Here鈥檚 a recap of the process:
  • Start with the exponential form \( w^{x+2} = r \).
  • Convert to logarithmic form \( \text{log}_w(r) = x+2 \).
  • Check if further simplification is needed.
Simplification is crucial because it helps us solve and understand the core principle behind the equation.

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