Problem 38 - Algebra A Combined Function

Question

Find the value of each logarithmic expression. $$ \log _{9} 9 $$

Step-by-Step Solution

Answer

The value of \( \log_{9} 9 \) is 1.

Step 1: Identify the Logarithmic Expression

The given expression is \( \log_{9} 9 \). This represents the logarithm of 9 to the base 9.

Step 2: Recall the Logarithm Rule for Identical Base and Argument

The rule for logarithms states that \( \log_{b} b = 1 \). This means that if the base of the logarithm and the argument (the number you're taking the logarithm of) are the same, then the value of the logarithm is 1.

Step 3: Apply the Rule to the Expression

Since both the base and the argument in \( \log_{9} 9 \) are 9, we can directly apply the rule to find that \( \log_{9} 9 = 1 \).

Key Concepts

Logarithm Properties

Understanding the properties of logarithms is crucial when solving logarithmic expressions. These properties help simplify the expressions and solve equations efficiently. One of the fundamental properties is the identity property, which states that if the base and the argument are the same, the logarithm equals 1.
Logarithms are the inverse operations of exponentiation. This means that they "undo" exponential equations, allowing us to find the exponent that a base has to be raised to, to obtain a specific number. The magical property here is that \(\log_{b}(b) = 1\).
Here are some common properties of logarithms to remember:

• Product Property: \(\log_{b}(xy) = \log_{b}(x) + \log_{b}(y)\)
• Quotient Property: \(\log_{b}\left(\frac{x}{y}\right) = \log_{b}(x) - \log_{b}(y)\)
• Power Property: \(\log_{b}(x^n) = n\cdot\log_{b}(x)\)

Remembering these properties can make dealing with logarithmic expressions much easier and less intimidating.

Base and Argument

In logarithmic expressions, the base and argument are the two primary components. The base is the number that is raised to a power to yield the argument. When you see a logarithm written as \(\log_{b}(x)\), \(b\) is the base, and \(x\) is the argument.
For example, in the expression \(\log_{9}(9)\), both the base and the argument are 9. This is a straightforward example of the identity property. Since the base and argument are the same, the logarithmic expression is easily evaluated as 1.

Knowing the components clearly helps in applying the correct logarithmic rules. Make sure to distinguish between the two, as they form the foundation of solving the equations.

• The Base: The number being raised to a power.
• The Argument: The number you achieve from raising the base to the power.

Solving Logarithmic Expressions

When it comes to solving logarithmic expressions, applying the correct properties and understanding the base and argument are vital. Most of the time, simplifying these expressions involves applying known properties to transform the expression into a simpler form.
To solve an expression like \(\log_{9}(9)\), you identify it immediately due to the properties: both the base and the argument are the same. Hence, the expression resolves directly to 1. This showcases the efficiency of logarithmic properties.
If other expressions involve different bases or arguments, the properties help you break them down:

• Using the product, quotient, or power properties to expand, condense, or equate they might offer multiple ways to reach the final answer.
• Change of Base Formula can also be used when the unlike base hinders direct simplification: \(\log_{b}(x) = \frac{\log_{c}(x)}{\log_{c}(b)}\), where \(c\) is a common base, usually 10 or \(e\).

With practice, these expressions can be solved swiftly, enhancing both accuracy and understanding of the nature of logarithms.