Problem 49 - College Algebra and Calculus: An Applied Approach

Question

Approximate the logarithm using the properties of logarithms, given \(\log _{b} 2 \approx 0.3562\), \(\log _{b} 3 \approx 0.5646\), and \(\log _{b} 5 \approx 0.8271$$\log _{b} \sqrt[3]{4 b}\)

Step-by-Step Solution

Answer

The approximate value for \( \log_b \sqrt[3]{4 b} \) is 0.7124.

Step 1: Express the Cube Root

Start by expressing the cube root in the argument of the logarithm as a power i.e. \( \log_b 4b^{1/3} \). The cube root of \( b \) is equivalent to \( b^{1/3} \). This allows the use of logarithmic power rule in the next step.

Step 2: Applying Logarithmic Power Rule

Implement the logarithmic power rule, which states that \( \log_b x^n = n \cdot \log_b x \). Applying to our problem translates the equation into \( (1/3) \cdot \log_b b + \log_b 4 \).

Step 3: Simplify Logarithmic Expression

The logarithm of any number to its own base equals 1. Thus, the term \( (1/3) \cdot \log_b b \) simplifies to \( 1/3 \). Furthermore, \( \log_b 4 \) can be written as \( 2 \cdot \log_b 2 \) (since 4 equals to \( 2^2 \)). Hence, the simplification of the expression leads to \( 1/3 + 2 \cdot \log_b 2 \).

Step 4: Substitute The Approximate Value For \(\log_b 2\)

Given in the problem, \( \log_b 2 \approx 0.3562 \). Substituting this into the resulting expression from the previous step gives the final approximate value for \( \log_b \sqrt[3]{4 b} \), which is \( 1/3 + 2 \cdot 0.3562 \).

Key Concepts

Logarithmic Power Rule

Understanding the logarithmic power rule is essential for manipulating complex logarithmic expressions. This rule states that if you have a log expression in the form of \( \log_b(x^n) \), you can simplify it to \( n \cdot \log_b(x) \).

• This is incredibly helpful because it lets you remove the exponent, which simplifies the expression considerably.
• Rather than dealing directly with roots, powers, or more complex scenarios, the logarithmic power rule breaks the expression into simpler, easily manageable parts.

Let's take an example from the exercise: \( \log_b 4b^{1/3} \) can utilize the power rule to result in \( (1/3) \cdot \log_b b + \log_b 4 \). This transformation highlights the utility of the rule, making the expression much easier to work with.

Approximation of Logarithms

Logarithms are often given with approximate values rather than precise definitions, especially for computational or practical purposes. Approximations help in swiftly estimating outcomes without calculating to an unnecessary level of precision. In the exercise, you have approximate values for \( \log_b 2 \), \( \log_b 3 \), and \( \log_b 5 \):

• \( \log_b 2 \approx 0.3562 \)
• \( \log_b 3 \approx 0.5646 \)
• \( \log_b 5 \approx 0.8271 \)

These approximations are particularly useful in finding or verifying solutions to logarithmic problems, such as ensuring that our calculated expression \( 1/3 + 2 \cdot \log_b 2 \) provides a viable approximation for \( \log_b \sqrt[3]{4b} \).

Simplification of Logarithmic Expressions

Simplifying logarithmic expressions makes them easier to solve or estimate. The process often involves using various logarithmic properties, such as the logarithmic power rule or known values.

• Firstly, converting a cube root or any root into an exponent form can simplify initial steps using the power rule.
• The next essential technique involves replacing terms with their approximations; for instance, rewriting \( \log_b 4 \) as \( 2 \cdot \log_b 2 \).
• This allows for direct substitution of known values, leading to easier calculations.

For this reason, the final step in the solution involved substituting approximate values for easy computation, leading to the resultant value of the expression that originally seemed complex.