Problem 53 - Intermediate Algebra

Question

For Problems \(51-59\), you are given \(\log _{2} 5=2.3219\) and \(\log _{2} 7=2.8074\). Evaluate each expression using Properties 11.5-11.7. $$ \log _{2} 125 $$

Step-by-Step Solution

Answer

\( \log_{2} 125 = 6.9657 \)

Step 1: Recognize the form

The given expression is \( \log_{2} 125 \). We need to express 125 in terms of powers that we have logarithmic information for. Recognize that 125 is a power of 5, specifically \( 125 = 5^3 \).

Step 2: Apply the power rule of logarithms

The power rule of logarithms states that \( \log_b (a^n) = n \cdot \log_b a \). Apply this to the expression \( \log_{2} 125 = \log_{2} (5^3) \). Thus, it becomes \( 3 \cdot \log_{2} 5 \).

Step 3: Substitute given logarithm value

We know that \( \log _{2} 5 = 2.3219 \) from the problem statement. Substitute this value into the equation: \( \log_{2} 125 = 3 \times 2.3219 \).

Step 4: Calculate the result

Calculate the product from Step 3: \( 3 \times 2.3219 = 6.9657 \). Therefore, \( \log_{2} 125 = 6.9657 \).

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a fundamental concept in the study of logarithms. This rule simplifies the process of dealing with exponents inside logarithmic expressions. It states that if you have a logarithm of a number raised to a power, you can bring the exponent in front of the log as a multiplier. The formula is expressed as follows:

\( \log_b (a^n) = n \cdot \log_b a \)

For example, consider \( \log_2 (5^3) \). According to the power rule, this expression can be rewritten as \( 3 \cdot \log_2 5 \). This rule helps by reducing the complexity of calculations, especially when evaluating logarithms using known values. By utilizing this rule, you can solve expressions more efficiently.

Logarithmic Expressions

Logarithmic expressions are mathematical statements that involve logarithms, which are the inverse operations of exponentiation. Logarithms help to determine the power to which a base must be raised to obtain a certain number. When working with logarithmic expressions, it is important to identify common properties that could simplify calculations.

For instance, expressions such as \( \log_2 125 \) require us to express 125 in terms of factors for which we have known logarithm values. In the given problem, 125 can be expressed as \( 5^3 \). This conversion helps us apply rules like the power rule to simplify the expression. Logarithmic expressions are crucial in solving complex equations and are widely used in scientific areas like chemistry and physics to handle large-scale calculations involving exponential growth or decay.

Evaluating Logarithms

Evaluating logarithms means finding the numerical value of a logarithmic expression using known values and logarithmic properties. To evaluate \( \log_2 125 \), we used the power rule of logarithms and given numerical values like \( \log_2 5 = 2.3219 \).

**Steps involved in evaluating logarithms:**

• Identify the base and the number you are dealing with (in this example, base 2 and 125).
• Express the number in terms of known logarithm values (125 is \( 5^3 \)).
• Apply logarithmic properties such as the power rule.
• Substitute known values and compute the result: \( 3 \cdot 2.3219 = 6.9657 \).

By following these steps, you can methodically evaluate logarithmic expressions to find precise answers. Understanding and applying these techniques is essential in mathematics, particularly when working with logarithms in various problem-solving scenarios.