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Chapter 4: Problem 75

Write each equation as an equivalent logarithmic equation. $$y=a^{z}$$

Short Answer

Expert verified
The equivalent logarithmic equation is: \[ z = \text{log}_a(y) \].

Step by step solution

01

Understand the Exponential Equation

The given equation is in exponential form: \[ y = a^{z} \]. We need to convert this into its equivalent logarithmic form.
02

Apply the Definition of a Logarithm

To rewrite the exponential equation as a logarithmic equation, use the definition of logarithms. The general form of an exponential equation is \[ y = a^{z} \], which can be rewritten as \[ z = \text{log}_a(y) \].
03

Rewrite the Equation

By applying the definition, the equivalent logarithmic equation becomes: \[ z = \text{log}_a(y) \].

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Key Concepts

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

exponential form
When dealing with logarithmic equations, understanding exponential form is crucial.
Exponential form is an expression where a base number is raised to an exponent to produce a value.
In our problem, the equation is given in exponential form: \(y = a^{z}\). Here:
  • \textbf{y} is the result or output.
  • \textbf{a} is the base.
  • \textbf{z} is the exponent.
To solve or manipulate these sorts of equations, it's essential to know how to switch to a logarithmic form.
definition of logarithms
To transition from exponential form to logarithmic form, grasping the definition of logarithms is key.
A logarithm answers the question: to what exponent must the base be raised, to produce a certain number?
Formally, if we have the exponential equation of the form \(y = a^{z}\), the logarithmic form is derived by asking:
  • 'What power should we raise \textbf{a} to, in order to get \textbf{y}?'
    This is written as:
\(z = \text{log}_a(y)\).
This equation states that \textbf{z} is the exponent to which the base \textbf{a} must be raised to obtain \textbf{y}.
Notice, a logarithm essentially undoes the action of an exponent.
logarithmic form conversion
Converting from exponential form to logarithmic form can seem tricky, but it's straightforward with practice.
Let's convert \(y = a^{z}\) into logarithmic form step-by-step.
  • Recognize the given form is exponential: \(y = a^{z}\)
  • Use the definition of logarithms to rewrite it, which means we need to find the exponent, \textbf{z}.
  • Logarithmic form states: \(z = \text{log}_a(y)\)
Now, you have successfully converted the given exponential equation into its equivalent logarithmic equation.
This method is valuable in solving problems where the exponent needs to be isolated or found.

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Most popular questions from this chapter

Depreciation and Inflation Boris won a \(\$ 35,000\) luxury car on Wheel of Fortune. He plans to keep it until he can trade it evenly for a new compact car that currently costs \(\$ 10,000\). If the value of the luxury car decreases by \(8 \%\) each year and the cost of the compact car increases by \(5 \%\) each year, then in how many years will he be able to make the trade?

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Solve \(\log _{2}(x-3)=8.\)
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