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}\left(3 b^{2}\right)\)

Short answer

The solution is 2.5646

Step-by-step solution

Expressing the Target Expression in terms of Given Logs

Using the property of logarithms, \(\log_b{MN} = \log_b{M} + \log_b{N}\), the logarithm \(\log _{b}\left(3 b^{2}\right)\) can be separated into \(\log_{b}3 + 2\log_{b}b\).

Simplifying using the Logarithm's Property

The \(\log _{b} b\) is equal to 1, since any log base 'a' of 'a' is equal to 1. So our equation simplifies to \(\log _{b} 3 + 2\).

Substitute the Value of the Given Logarithm

The approximation value given for \(\log _{b} 3\) in the problem is 0.5646. By substituting this value into the equation from Step 2, it can be determined the value of the expression: \(0.5646 + 2\).

Key concepts

Logarithm Approximation

When dealing with complex mathematical problems, we often encounter situations where we need to approximate the value of a logarithmic expression. Logarithm approximation is a useful technique for estimating the value of a logarithm when the exact value is either unknown or difficult to calculate directly.

For example, in the given exercise, we have the values of \( \log_b{2}, \log_b{3}, \text{and } \log_b{5} \) approximated to three decimal places. Using these approximate values, we can estimate the logarithm of other expressions as long as we express them in terms of the known base-\(b\) logarithms. This method is particularly valuable when dealing with non-standard numbers or when a calculator with logarithm capabilities isn't available.

By understanding that logarithms are exponents and representing various quantities as powers of a base, we can use rounding and other approximations to gain a reasonable estimate of the logarithms, which is often sufficient for practical purposes. In the digital age, logarithm approximation also serves as the basis for various numerical methods and computer algorithms that require quick estimations rather than precise values.

Calculating Logarithms

Calculating logarithms is a fundamental operation in algebra and is frequently used in various fields such as science, engineering, and finance. To calculate a logarithm, you want to find the power to which a particular base must be raised to obtain a given number. The general form for a logarithm is expressed as \( \log_b{X} = Y \), meaning that \( b^Y = X \).

In the exercise provided, we calculate \( \log_b(3 b^{2}) \) by breaking it down into simpler parts for which we have approximate values or can easily evaluate. Understanding how to manipulate logarithmic expressions, such as splitting the logarithm of a product into a sum of logarithms, or recognizing the logarithm of a base to its own power equals 1, are crucial skills to solve such problems.

Moreover, familiarity with the laws of logarithms allows us to transition between logarithmic and exponential forms, enabling further manipulation and calculation of these expressions. While some logarithms, like base-10 or natural logarithms (\( \log_{10} \) or \( \ln \)), might be readily calculated using scientific calculators, understanding the principles behind these calculations is essential for more complex or unconventional problems where direct computation isn't possible.

Logarithmic Properties

The properties of logarithms are mathematical rules that help simplify the process of calculation and manipulation of logarithmic expressions. These properties are based on fundamental characteristics of exponents as logarithms are inverse operations to exponentiation.

Several key properties of logarithms include:

• Product Rule: \( \log_b(MN) = \log_b(M) + \log_b(N) \), which states that the logarithm of a product is the sum of the logarithms of the factors.
• Quotient Rule: \( \log_b(\frac{M}{N}) = \log_b(M) - \log_b(N) \), which indicates that the logarithm of a quotient is the difference of the logarithms.
• Power Rule: \( \log_b(M^k) = k \cdot \log_b(M) \), suggesting that the logarithm of a number raised to a power is the power multiplied by the logarithm of the number.
• Change of Base Formula: \( \log_b(M) = \frac{\log_k(M)}{\log_k(b)} \), which allows calculation of logarithms with one base in terms of logarithms with another base.
• Base Identity: \( \log_b(b) = 1 \), since any number raised to the power of 1 is itself.

Applying these properties, as seen in the given exercise, greatly simplifies the process of calculating logarithms that may initially seem complicated. Through the understanding and proper application of these properties, even logarithm approximation becomes a more accessible concept, aiding in problem solving and analytics across various disciplines.