Problem 15 - Elementary and Intermediate Algebra

Question

a. Find \(\frac{\log 8}{\log 5} .\) Round to four decimal places. b. Find \(\frac{3 \ln 12}{\ln 4-\ln 2} .\) Round to four decimal places.

Step-by-Step Solution

Answer

a. 1.2920 b. 10.7547

Step 1: Understanding Logarithms and Change of Base Formula

To solve this problem, we'll use the change of base formula, which states that for any logarithm \( \log_b(a) \), it can be expressed as \( \frac{\log_k(a)}{\log_k(b)} \) for any positive base \( k \). In both these exercises, we'll use this property to rewrite and evaluate the expressions.

Step 2: Compute \( \frac{\log 8}{\log 5} \) Using Logarithm Tables

For part (a), we'll express both \( \log 8 \) and \( \log 5 \) using base 10 and calculate their values using a calculator. Calculate \( \log 8 \approx 0.9031 \) and \( \log 5 \approx 0.6990 \). The expression becomes \( \frac{0.9031}{0.6990} = 1.2920 \).

Step 3: Simplifying the Expression \( \ln 4 - \ln 2 \)

For part (b), we first simplify the expression \( \ln 4 - \ln 2 \) using the property \( \ln a - \ln b = \ln \left(\frac{a}{b}\right) \). So, it becomes \( \ln 2 \), because \( \frac{4}{2} = 2 \).

Step 4: Compute \( \frac{3 \ln 12}{\ln 2} \) Using Natural Logarithm

Next, we calculate \( \ln 12 \) and \( \ln 2 \) using a calculator. \( \ln 12 \approx 2.4849 \) and \( \ln 2 \approx 0.6931 \). Then compute \( \frac{3 \times 2.4849}{0.6931} \), which simplifies to \( \frac{7.4547}{0.6931} = 10.7547 \).

Step 5: Final Results for Both Parts

Rounding both parts to four decimal places, for part (a) \( \frac{\log 8}{\log 5} = 1.2920 \) and for part (b) \( \frac{3 \ln 12}{\ln 4-\ln 2} = 10.7547 \). Ensure all calculations maintain this precision.

Key Concepts

Change of Base Formula

Understanding how to change the base of a logarithm is incredibly useful for solving many logarithmic problems. The change of base formula allows us to express a logarithm in a different base, making calculations easier when we have a calculator that may only evaluate common logarithms (base 10) or natural logarithms (base \( e \)).

The formula is: if you have \( \log_b(a) \), you can rewrite it as:

• \( \frac{\log_k(a)}{\log_k(b)} \)

Here, \( k \) is any new positive base you choose. Typically, for calculators, \( k \) is 10 or \( e \) because these are common functions calculators can handle.

This formula is vital because it shows that the base doesn't affect the overall result — you just need consistency. For the problem given with \( \frac{\log 8}{\log 5} \), we use base 10 logarithms, also called "log," because this is standard on most calculators. Then, applying the formula allows us to evaluate even if the numbers involve different bases initially.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a special type of logarithm where the base is the irrational number \( e \), approximately equal to 2.71828. It's particularly important in calculus and many mathematical applications due to its unique properties and relationship with exponential functions.

Natural logarithms simplify many problems, especially those involving growth models, compound interest, or decay, because they convert multiplicative processes into additive ones. This makes complicated multiplication problems easier to handle.

In the exercise, you saw \( \ln \) being used. For instance, \( \ln 4 - \ln 2 \) was simplified to \( \ln(\frac{4}{2}) \) or \( \ln 2 \). This simplification is due to a logarithmic property that allows subtraction to become division inside the log function. Calculators typically have a specific button for \( \ln \) because of its frequent use, especially in advanced math topics.

Logarithm Properties

Logarithms have some essential properties that make them powerful tools in mathematics. Let's explore some fundamental properties:

• Product Property: \( \log_b(MN) = \log_b(M) + \log_b(N) \)
• Quotient Property: \( \log_b(\frac{M}{N}) = \log_b(M) - \log_b(N) \)
• Power Property: \( \log_b(M^p) = p \cdot \ln_b(M) \)

These properties help in simplifying complex expressions. For instance, the exercise used the quotient property \( \ln 4 - \ln 2 \) to simplify "inside" the log, turning it into a straightforward division. This quick manipulation reduces complexity in equations involving logs.

Logarithms turn multiplication into addition and division into subtraction. \( \ln \) and other logs are crucial in fields like engineering, physics, and any area involving growth or change rates. They allow problem-solving involving multi-step arithmetic operations to be more manageable, showcasing their versatility and utility across various disciplines.