Problem 29 Simplify. $$ \log _{9} 3 $... [FREE SOLUTION]

Chapter 12: Problem 29

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

Short Answer

Expert verified

$ \log_{9} 3 = \frac{1}{2} $

Step by step solution

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 $9^{x} = 3$.

Convert the Base

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

Simplify the Exponent

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

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 $2x = 1$.

Solve for x

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

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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91影视

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

Short Answer

Step by step solution

Understand the Logarithm Definition

Convert the Base

Simplify the Exponent

Equate the Exponents

Solve for x

Interpret the Result

Key Concepts

Logarithm Definition

Exponent Rules

Simplifying Logarithms

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