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Chapter 10: Problem 18

Write in exponential form. $$\log _{2} 512=9$$

Short Answer

Expert verified
The exponential form is \(2^9 = 512\).

Step by step solution

01

Identify the logarithmic equation components

The given logarithmic equation is \(\text{log}_{2} 512 = 9\). Here, the base is 2, the argument is 512, and the result is 9.
02

Rewrite the logarithmic equation in exponential form

The equation \(\text{log}_{b} a = c\) can be rewritten in exponential form as \(b^c = a\). In this case, b = 2, c = 9, and a = 512.
03

Form the exponential equation

Replace the values with the identified components: \(2^9 = 512\).

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Key Concepts

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

Logarithmic Equation
A logarithmic equation has an expression in the form of \text{log}_b a = c\. In this expression:
  • \(b\) is the **base** of the logarithm
  • \(a\) is the **argument**
  • \(c\) is the **result**
Let's take the example \text{log}_2 512 = 9\. Here, 2 is the base, 512 is the argument, and 9 is the result. The purpose of the logarithmic equation is to find the exponent (result) to which the base must be raised to achieve the argument. In this case, 2 must be raised to the power of 9 to get 512.

Understanding how to identify these components is the first step to manipulating logarithmic equations correctly. Once you have these components, you can rewrite the logarithmic form into its equivalent exponential form.
Exponential Equation
An exponential equation involves expressions where a constant base is raised to a variable exponent. The general form is \text{b}^c = a\, where:
  • \(b\) is the **base**
  • \(c\) is the **exponent**
  • \(a\) is the **result**
For example, let's consider \text{2}^9 = 512\. This equation tells us that when 2 is multiplied by itself 9 times, it gives 512.
You can convert a logarithmic equation into an exponential form for easier calculation and understanding. For instance, given \text{log}_2 512 = 9\, you can write it in exponential form as \text{2}^9 = 512\.

This transformation from logarithmic form to exponential form helps in solving various equations and understanding the relationships between numbers.
Base and Argument
In both logarithmic and exponential equations, understanding the base and argument is crucial.
The **base** is the number that gets multiplied by itself. In the equation \text{log}_2 512 = 9\, the base is 2.
The **argument**, on the other hand, is the result we aim to achieve through the exponentiation process. For example, in \text{log}_2 512 = 9\, the argument is 512.

Here's a breakdown of these terms in context:
  • Logarithmic Equation: \text{log}_b a = c\, where \(b\) is the base, \(a\) is the argument
  • Exponential Equation: \text{b}^c = a\, where \(b\) is the base to the power of \(c\) resulting in \(a\)
Understanding the roles of the base and the argument will make these equations and their solutions much clearer. So, always identify these components first before proceeding with transformations or solving the equations.

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Most popular questions from this chapter

Solve each equation. Give exact solutions. $$ \log _{3} x+\log _{3}(2 x+5)=1 $$

Solve each equation. Approximate solutions to three decimal places. $$ 2^{x+3}=5^{x} $$

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. $$ \log _{3} \sqrt{2} $$

The population of deer (in thousands) in a certain area is approximated by the logarithmic function $$f(x)=\log _{5}(100 x-75)$$where \(x\) is the number of years since 2017 . During what year is the population expected to be 4 thousand deer?

Solve each equation. Approximate solutions to three decimal places. $$ 9^{-x+2}=13 $$
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