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Chapter 12: Problem 29

Simplify. $$ \log _{9} 3 $$

Short Answer

Expert verified
\( \log_{9} 3 = \frac{1}{2} \)

Step by step solution

01

Understand the Logarithm Definition

The expression \(\log_{9} 3\) asks for the exponent to which 9 must be raised to yield 3. In other words, if \(x = \log_{9} 3\), then \(9^{x} = 3\).
02

Convert the Base

Recall that \9 = 3^2\. So, rewrite the base 9 in terms of 3: \(9^x = (3^2)^x\).
03

Simplify the Exponent

Use the power of a power rule: \( (a^{m})^{n} = a^{mn} \). Thus, \( (3^2)^x = 3^{2x} \).
04

Equate the Exponents

Set \(3^{2x}\) equal to \(3\), so \(3^{2x} = 3^1\). Since the bases are the same, the exponents must be equal, hence \(2x = 1\).
05

Solve for x

Solve the equation \(2x = 1\) to find \(x\). Divide both sides by 2: \(x = \frac{1}{2} \).
06

Interpret the Result

Thus, the simplified form of \(\log_{9} 3 \) is \( \frac{1}{2} \).

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Key Concepts

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

Logarithm Definition
A logarithm answers the question: To what power must a given base be raised, to get a certain number? When you see something like \(\log_{b}a\), it asks: 'To what power must the base \(b\) be raised to yield \(a\)?' Let's break that down a bit more:
  • In \(\log_{9} 3\), the base is 9.
  • The answer you get is the exponent to which you raise 9 to get 3.
So, if
\[ x = \log_{9} 3 \]
then
\[ 9^{x} = 3 \] This is what forms the basis for working with logarithms!
Exponent Rules
Exponents play a crucial role in simplifying logarithms. Here are some basic exponent rules that come in handy:
  • \((a^m)^n = a^{mn}\) 鈥� Power of a power: Multiply the exponents.
  • \a^1 = a\ 鈥� Any number to the power of 1 is the number itself.
Let's see how exponent rules help in our example:
Step 2: Convert the Base
Since \(9 = 3^2\), rewrite the base 9 as \(3^2\):
\[ 9^x = (3^2)^x \]
Step 3: Simplify the Exponent
Use power of a power rule: \((a^m)^n = a^{mn}\). Thus,
\[ (3^2)^x = 3^{2x} \]
Simplifying Logarithms
Simplifying logarithms often involves converting bases and applying exponent rules. Let's go through our example:
\[\log_{9} 3\]
Step 4: Equate the Exponents
Since \( 3^{2x} = 3^1 \), equate the exponents:
\[ 2x = 1 \]
Step 5: Solve for x
Divide both sides by 2:
\[ x = \frac{1}{2}\]
Step 6: Interpret the Result
So, the simplified form of
\[ \log_{9} 3 \] is \[ \frac{1}{2} \]
To recap:
  • Rewrite the base to express it in simpler terms.
  • Apply exponent rules to simplify.
  • Equate and solve for the exponent to get your simplified logarithm.

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Most popular questions from this chapter

Review factoring polynomials (Sections \(6.1-6.6)\) Factor $$ 5 x^{4}-10 x^{3}-3 x^{2}+6 x $$

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Use the table below to find each value, if possible. $$\begin{array}{|c|c|c|} \hline {x} & {f(x)} & {g(x)} \\ \hline {1} & {0} & {1} \\ {2} & {3} & {5} \\ {3} & {2} & {8} \\ {4} & {6} & {5} \\ {5} & {4} & {1} \\ \hline \end{array}$$ $$g(f(5))$$

Simplify. $$ \log _{1 / 5} 25 $$
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