Problem 39 - Algebra A Combined Function

Question

Find the value of each logarithmic expression. $$ \log _{10} 100 $$

Step-by-Step Solution

Answer

The value of \( \log_{10} 100 \) is 2.

Step 1: Understand the Logarithm Definition

The logarithm \( \log_b a \) is the power to which the base \( b \) must be raised to equal \( a \). So, \( \log_{10} 100 \) asks, "What power must 10 be raised to, to get 100?"

Step 2: Express 100 as a Power of 10

Recognize that 100 can be written as \( 10^2 \). That is because 10 multiplied by itself equals 100.

Step 3: Apply the Logarithmic Property

Since 100 = \( 10^2 \), the expression \( \log_{10} 100 \) can be rewritten as \( \log_{10} (10^2) \).

Step 4: Use the Power Rule of Logarithms

The power rule of logarithms states that \( \log_b (b^c) = c \). Therefore, \( \log_{10} (10^2) = 2 \).

Step 5: Conclude the Evaluation

Following from the power rule, the value of \( \log_{10} 100 \) is 2.

Key Concepts

Logarithmic Expressions

Logarithmic expressions are mathematical statements that involve the logarithm function. Logarithms can be seen as the inverse operations of exponentiation. When you deal with logarithmic expressions, you are essentially asking the question: "To what power must the base be raised to produce a given number?" In a logarithmic expression like \( \log_{b} a \), \( b \) is called the base and \( a \) is the argument.

• For example, in \( \log_{10} 100 \), the base is 10, and the argument is 100.
• The result of \( \log_{10} 100 \) yields the exponent that 10 must be raised to, in order to yield 100.

Logarithmic expressions are commonly used in numerous fields such as science, engineering, and even music to manage exponential growth and decay.

Base 10 Logarithms

Base 10 logarithms, also known as common logarithms, are widely used in mathematics. The base 10 logarithm, denoted as \( \log_{10} \) or simply \( \log \), is particularly useful because it mirrors our decimal number system. When working with base 10 logs, you often deal with quantities that involve powers of 10, making it easier to calculate and understand.

• For instance, calculating \( \log_{10} 100 \) means determining how many times you multiply 10 to get 100.
• The calculation reveals that \( 10^2 = 100 \), so \( \log_{10} 100 = 2 \).

Understanding base 10 logarithms helps simplify complex calculations by transforming multiplication into addition, which is much more intuitive when performing large-scale computations.

Power Rule of Logarithms

The power rule of logarithms is an important property that can simplify evaluating logarithmic expressions. The power rule states that \( \log_b (b^c) = c \). This implies that if you have a base raised to a power, you can simply "bring down" the exponent as the result of the logarithm. In cases where the base and the argument's base match, this rule makes calculations straightforward and efficient.

• For example, with \( \log_{10} (10^2) \), we directly obtain 2 because the base (10) and the argument's base (10) are the same.
• This rule is particularly useful in simplifying expressions and solving equations, especially in algebra and calculus.

By mastering the power rule, you can better tackle logarithmic problems and apply these concepts to real-world scenarios more effectively.