Problem 89 - Intermediate Algebra

Question

Assume that \(\log 4 \approx 0.6021, \log 7 \approx 0.8451,\) and \(\log 9 \approx 0.9542 .\) Use these values to evaluate each logarithm. See Example 8 . $$ \log _{b} \frac{4}{63} $$

Step-by-Step Solution

Answer

\(\log_b \frac{4}{63} \approx -1.1972\)

Step 1: Recall the Logarithm Division Rule

The logarithm of a quotient can be expressed as the difference of the logarithms of the numerator and the denominator. Therefore, for the expression \(\log_b \frac{4}{63}\), we have \(\log_b \frac{4}{63} = \log_b 4 - \log_b 63\).

Step 2: Break Down the Denominator

We need to express the denominator 63 in terms of factors for which we know the approximate logarithm values. Notice that \(63 = 7 \times 9\). Thus, \(\log_b 63\) can be expressed as \(\log_b 7 + \log_b 9\).

Step 3: Substitute Known Logarithm Values

Substitute the given logarithm values into the formula: \(\log_b 4 - \log_b (7 + 9)\). This results in: \[\log_b \frac{4}{63} = \log_b 4 - (\log_b 7 + \log_b 9)\].

Step 4: Calculate the Expression with Given Values

Plug in the known logarithm values to compute: \[\log_b \frac{4}{63} = 0.6021 - (0.8451 + 0.9542)\].

Step 5: Perform the Arithmetic Operations

First, calculate the sum of the denominators' logarithms: \(0.8451 + 0.9542 = 1.7993\). Then find the difference: \(0.6021 - 1.7993 = -1.1972\). Thus, \(\log_b \frac{4}{63} \approx -1.1972\).

Key Concepts

Properties of Logarithms

Logarithms are fascinating mathematical tools used to simplify complex calculations. The core properties of logarithms allow us to manipulate and transform logarithmic expressions in various ways:

• Product Property: The logarithm of a product is the sum of the logarithms. Specifically, \[ \log_b (xy) = \log_b x + \log_b y \]
• Quotient Property: The logarithm of a quotient is the difference between the logarithms. For example,\[ \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \]
• Power Property: The logarithm of a power is the exponent times the logarithm of the base. Which is expressed as\[ \log_b (x^n) = n \cdot \log_b x \]

These identities are vital to simplifying log expressions, especially in calculations involving multiple steps. They streamline problem-solving, making complex expressions manageable. In our exercise, we primarily used the quotient property to break down and solve the logarithmic expression.

Logarithm Division Rule

The logarithm division rule, also known as the quotient property, is a powerful tool for simplifying division inside a logarithm. It states that the logarithm of a quotient is simply the difference between the logarithms of the numerator and the denominator: \[ \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \] For instance, in the initial problem, we had \[ \log_b \left( \frac{4}{63} \right) \] By applying the rule, this expression is transformed into \[ \log_b 4 - \log_b 63 \] This transformation is very useful because it allows us to work with simpler numbers. Instead of evaluating the logarithm of a fraction directly, breaking it down into parts makes it easier to substitute known values for further calculation. This rule significantly aids in simplifying problems involving division within logarithms.

Logarithmic Calculations

Logarithmic calculations involve evaluating complex expressions more conveniently by applying properties of logarithms. Let's walk through how the given problem was solved using known logarithmic values:First, the expression \[ \log_b \left( \frac{4}{63} \right) \] was broken down using properties into \[ \log_b 4 - (\log_b 7 + \log_b 9) \] The given values for these logarithms were:

• \(\log_b 4 = 0.6021\)
• \(\log_b 7 = 0.8451\)
• \(\log_b 9 = 0.9542\)

Next, substituting these known values, the expression became \[ 0.6021 - (0.8451 + 0.9542) \] Perform the arithmetic calculation:

• First, add \(0.8451 + 0.9542\) to get \(1.7993\)
• Then, compute \(0.6021 - 1.7993\) to obtain \(-1.1972\)

Thus, the answer \[ \log_b \left( \frac{4}{63} \right) \approx -1.1972 \] This approach efficiently uses arithmetic and logarithmic identities to streamline solving logarithmic expressions.