Problem 69 - Algebra A Combined Function

Question

Evaluate each expression. $$ \log _{7} 7^{2} $$

Step-by-Step Solution

Answer

The value is 2.

Step 1: Understanding the Properties of Logarithms

The expression given is \( \log_{7} 7^{2} \). We need to use the property of logarithms that states \( \log_{b} (b^x) = x \). This property applies because the base of the log, 7, is the same as the base of the exponential, which is 7.

Step 2: Applying the Logarithm Property

Since \( \log_{7} 7^{2} \) matches the form \( \log_{b} (b^x) = x \), we can directly apply this property. This simplifies our expression to: \[ \log_{7} 7^{2} = 2 \]

Step 3: Verify the Solution

We can check our work by considering the inverse operation. If we raise 7 to the power of our result, we should retrieve the original argument inside the logarithm. That is, if \( 7^{2} = 49 \), then \( \log_{7} 49 = 2 \), confirming our solution is correct.

Key Concepts

Properties of Logarithms

Logarithms have several properties that make them an invaluable tool for simplifying calculations and solving equations. One essential property is the Power Rule, which was used in our original exercise. This states that when you have a logarithm where the base of the log is the same as the base of the power in the argument, the logarithm can be simplified. In math terms, this is shown as \( \log_{b} (b^x) = x \). Drop the log and power bases, leaving just the exponent.

• Power Rule: \( \log_{b} (b^x) = x \)

This rule turns complex exponential expressions into simple arithmetic, making calculations easier to handle.
Understanding other properties, like the Product Rule \( \log_{b}(MN) = \log_{b} M + \log_{b} N \) and the Quotient Rule \( \log_{b}\left(\frac{M}{N}\right) = \log_{b} M - \log_{b} N \), further enriches your ability to manipulate and solve log equations efficiently.

Exponents

Exponents play a crucial role in mathematics and in our specific exercise. When you see a number raised to a power, like \( 7^2 \), it represents a base number being multiplied by itself a certain number of times. In this instance, \( 7^2 = 7 \times 7 \).
Understanding the basic rules of exponents helps in dealing with algebraic expressions and transformations, especially in logarithms.

• Multiplying Powers with the Same Base: \( b^m \cdot b^n = b^{m+n} \)
• Power to a Power: \( (b^m)^n = b^{m \times n} \)
• Negative Exponent: \( b^{-n} = \frac{1}{b^n} \)

These rules allow you to simplify expressions and make them easier to solve. Becoming proficient with exponents also provides a solid foundation for working with logarithms, as they essentially reverse exponential calculations.

Logarithmic Expressions

Logarithmic expressions can seem complex at first, but recognizing their connection to exponents helps in simplifying them. In our exercise, we dealt with an expression of the form \( \log_{b} (b^x) \). The purpose of a logarithm is to determine the power or exponent to which a base number must be raised to obtain a given value.
This is what makes understanding logarithmic expressions so impactful:

• The basic definition: If \( b^x = n \), then \( \log_{b} n = x \).
• They simplify situations where you have unknown exponents.
• They assist in switching between exponential and multiplicative perspectives.

The more you practice, the more intuitive they become, especially when manipulating the equations using properties like Power, Product, and Quotient rules. Working through logarithmic expressions step by step leads to a deeper understanding and appreciation of their role in math and science.