Problem 46 - 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} 45\)

Step-by-Step Solution

Answer

The logarithm \( \log_{b} 45 \) is approximately 1.96

Step 1: Rewrite using properties of logarithms

45 can be written as a product of the numbers 5 and 9. 9 can further be written as a product of two 3s. So, \( \log_{b} 45 \) can be rewritten as \( \log_{b} 5 + 2 \cdot \log_{b} 3 \).

Step 2: Substitute given approximations

Substitute the approximations given in the problem for \( \log_{b} 2 \), \( \log_{b} 3 \), and \( \log_{b} 5 \) into the equation. This yields \( 0.8271 + 2 \cdot 0.5646 \).

Step 3: Solve the equation

Calculate the result of the equation. \( 0.8271 + 2 \cdot 0.5646 = 1.9563 \)

Step 4: Round off

Round off the result for practical purposes. The final answer will therefore be approximately \( 1.96 \)

Key Concepts

Properties of Logarithms

Logarithms have certain properties that make them extremely useful for simplifying complex calculations. These properties help break down logarithms into simpler parts, making them easier to calculate or approximate.

Some important properties include:

• Product Property: The logarithm of a product is the sum of the logarithms of its factors. Formally, this is expressed as \( \log_b(xy) = \log_b x + \log_b y \).
• Quotient Property: The logarithm of a quotient is the difference between the logarithms of the numerator and the denominator, represented as \( \log_b(\frac{x}{y}) = \log_b x - \log_b y \).
• Power Property: The logarithm of a power is the exponent times the logarithm of the base, which is written as \( \log_b(x^n) = n \cdot \log_b x \).

These properties are particularly useful in breaking down numbers into prime factors, which was demonstrated in our original exercise. By understanding and applying these properties, complex logarithmic equations can be converted into a series of simpler calculations.

Logarithmic Approximation

Logarithmic approximation involves estimating the value of a logarithm using given data or using simplifications. In many cases, exact values of logarithms are not as crucial as approximate values, which provide useful and quick estimations. This is especially true in applications like scientific calculations or computer algorithms where speed and efficiency are key.

In our case, we approximated \( \log_{b} 2 \), \( \log_{b} 3 \), and \( \log_{b} 5 \) as 0.3562, 0.5646, and 0.8271 respectively. By substituting these values in the properties of logarithms, we derived \( \log_{b} 45 \). This allowed us to get an approximate value of 1.96 for \( \log_{b} 45 \) quickly.

Approximations remain crucial when dealing with logarithms, as there are often no simple expressions for logarithms of arbitrary numbers. Thus, quick approximations save time and maintain acceptable accuracy for most practical applications.

Mathematical Expressions

Crafting mathematical expressions is a foundational skill in mathematics, essential for solving problems and communicating complex ideas. A mathematical expression is a combination of numbers, variables, operations, and other mathematical symbols arranged in a logical manner to represent a concept or a relationship. They are used to express everything from basic arithmetic operations to more intricate algebraic formulas.

In the exercise, starting from the expression for \( \log_{b} 45 \), we utilized the properties of logarithms to rewrite and simplify it into \( \log_{b} 5 + 2 \cdot \log_{b} 3 \). Such manipulations often involve expressing complex numbers in terms of their prime factors or simpler numbers.

Understanding how to express one number in terms of others is very useful in simplifying big calculations. This skill supports the reduction of errors and can make it easier to recognize patterns or known values that aid in solving problems more efficiently. Learning to craft and simplify mathematical expressions opens doors to various computational techniques, critical in advanced fields of math and science.