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Chapter 4: Problem 20

Write the equation in exponential form. $$ \log _{9} 81=2 $$

Short Answer

Expert verified
9^2 = 81

Step by step solution

01

Identify the components of the logarithmic equation

In the given equation \( \log _{9} 81=2 \), identify the base (9), the argument (81), and the result (2).
02

Recall the definition of a logarithm

Recall that the definition of \( \log _{b} a = c \) can be rewritten in exponential form as \( b^c = a \).
03

Rewrite the equation using the exponential form

Using the identified components and the definition, rewrite \( \log _{9} 81=2 \) as \( 9^2 = 81 \).

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Key Concepts

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

Understanding Logarithms
A logarithm is a mathematical way to express something that is often written in exponential form. In essence, a logarithm answers the question: 'To what power must we raise a certain number, known as the base, to get another number?' This is written as: \( \log_b{a}=c \), where \( b \) is the base, \( a \) is the argument, and \( c \) is the result. For example, in the equation \( \log _{9} 81=2 \), 9 is the base, 81 is the argument, and 2 is the result. To fully grasp logarithms, always recall they relate to their exponential form.
How to Transform Logarithms to Exponential Form
Transforming a logarithmic equation to its exponential equivalent is simpler than it sounds. The definition \( \log_b{a}=c \) translates to \( b^c = a \) in exponential form. This means that the logarithm tells us how many times we need to multiply the base by itself to get the argument. For the given equation \( \log _{9} 81=2 \), we rewrite it in exponential form as \ 9^2 = 81 \. Here, \( b \) (9) raised to the power \( c \) (2) equals \ a \ (81). Breaking it down:
  • Base (b) = 9
  • Exponent (c) = 2
  • Argument (a) = 81
The logical flow from logarithmic to exponential form is simply taking the statement 'logarithm' and visualizing it as a power equation.
Understanding the Base in Logarithmic Equations
The base in a logarithmic equation is the number that is raised to a certain power. In the example \( \log _{9} 81=2 \), the base is 9. The importance of the base is foundational because it determines how we scale the powers to match the argument. Different bases change the exponential growth rate. For instance:
  • Base 10 is common in scientific notation.
  • Base 2 is often used in computer science (binary).
  • Base 9, like in our example, is less common but operates the same way.
Knowing the base helps you understand the relationship between the logarithmic and exponential forms in any given problem.

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Most popular questions from this chapter

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \log _{4}(3 d)+\log _{4} 1=\log _{4}(3 d) $$

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(10^{3+4 x}-8100=120,000\)

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(e^{2 x}-9 e^{x}-22=0\)

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log \left(x^{2}+7 x\right)=\log 18\)

Compare the graphs of the functions. $$ Y_{1}=\ln (2 x) \quad \text { and } \quad Y_{2}=\ln 2+\ln x $$
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