Problem 41 - Algebra A Combined Function

Question

Find the value of each logarithmic expression. $$ \log _{3} 81 $$

Step-by-Step Solution

Answer

The value is 4.

Step 1: Identify the Base and Argument

The logarithmic expression is \( \log_{3} 81 \). Here, the base of the logarithm is 3, and the argument is 81. The expression asks for the exponent you would raise the base, 3, to in order to get 81.

Step 2: Convert to Exponential Form

Remember that \( \log_{b}(x) = y \) implies \( b^y = x \). With \( \log_{3} 81 \), this becomes \( 3^y = 81 \).

Step 3: Find the Exponent

Determine the power of 3 that results in 81. We know that: \[3^1 = 3, \ 3^2 = 9, \ 3^3 = 27, \ 3^4 = 81\] Hence, \( y = 4 \).

Step 4: Verify the Calculation

Check that \( 3^4 = 81 \) is a true statement. Calculate as necessary: \( 3^4 = 3 \times 3 \times 3 \times 3 = 81 \). Since it’s correct, the logarithmic expression was solved accurately.

Key Concepts

Exponential Form

When we talk about exponential form, we are essentially describing how numbers can be expressed as powers. In mathematics, this concept is fundamental and highly useful for simplifying and solving equations. For instance, any number can be expressed as a product of repeated multiplication, such as:

• \( 3^4 \) represents 3 multiplied by itself four times: \( 3 \times 3 \times 3 \times 3 \).
• The result of this operation is 81, which means that 81 can be written in exponential form as \( 3^4 \).

Understanding the exponential form helps us reverse the process, which is the essence of logarithms. Being able to transition smoothly between exponential and logarithmic forms allows for easier solving of math problems involving exponents.

Base of a Logarithm

In a logarithmic expression, the base is the number you repeatedly multiply. It is the number raised to a power. In the expression \( \log_{3} 81 \), 3 is the base. Understanding the base of a logarithm is crucial as it dictates the factor that is repeatedly used in multiplication.

• For example, if the base is 3, you are working with multiples of 3.
• The base in logarithms is the same number that you use as the multiplier in the corresponding exponential form.

Knowing how to identify and work with the logarithm's base is vital for solving logarithmic equations.
The base sets the stage for determining the exponent, a key component when interpreting the equation.

Finding Exponents

Finding exponents involves determining the power to which a base number must be raised to reach a certain value. In the context of logarithms, this means solving according to the form \( b^y = x \).

• For the logarithmic expression \( \log_{3} 81 \), you need to find \( y \) such that \( 3^y = 81 \).
• By breaking down calculations: \( 3^1 = 3, \ 3^2 = 9, \ 3^3 = 27, \ 3^4 = 81 \), you discover that \( y = 4 \).

This shows you the exponent of 4 is necessary for 3 to reach 81. Finding exponents is a process of matching multiplicative combinations to identify the exact power needed, which is a foundational skill in algebra.