Problem 42 - 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} 81\)

Step-by-Step Solution

Answer

Substituting the known value of \( \log_b{3} \) in the equation from step 2, we get \( 4\log_b{3} = 4 \times 0.5646 = 2.2584 \). So, \( \log_b{81} \approx 2.2584 \).

Step 1: Break down the number 81

Recognize that 81 is a power of 3, specifically \(3^4\). This initial observation will play a crucial role in solving this exercise.

Step 2: Apply the power rule of logarithms

The power rule of logarithms allows a power in a logarithm's argument to be moved out front as a multiplier. So, rewrite \( \log_b{81} \) as \( \log_b{3^4} \). Then, using the power rule, it becomes \( 4\log_b{3} \).

Step 3: Substitute the known value

You can substitute the known value of \( \log_b{3} \) from the given in the exercise, which is 0.5646. Therefore, \( 4\log_b{3} = 4 \times 0.5646 \).

Key Concepts

Properties of Logarithms

Logarithms have several properties that make them incredibly useful in simplifying complex mathematical expressions. These properties help in the manipulation and calculation of logarithmic expressions without directly using a calculator or a computational tool. Here are the key properties you should know:

• Product Rule: This rule states that the logarithm of a product is the sum of the logarithms of the factors. Mathematically, it can be expressed as: \[\log_b (MN) = \log_b M + \log_b N\]
• Quotient Rule: This rule says that the logarithm of a quotient can be simplified to the difference of the logarithms of the numerator and the denominator:\[\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N\]
• Power Rule: This property allows you to bring down the power in the argument as a multiplier:\[\log_b (M^n) = n\cdot\log_b M\]

These properties are instrumental in breaking down more complex logarithmic expressions into something more manageable, as seen in the original exercise with \(\log_b 81\). Understanding these rules is crucial for anyone studying mathematics, especially when dealing with exponential growth, sound intensity, and other real-world applications.

Power Rule of Logarithms

One of the most potent tools in logarithmic calculations is the power rule. This rule is particularly useful when dealing with logarithms whose arguments are exponential in nature. It's straightforward and can be applied anytime a logarithmic function contains an exponent within its argument.

To apply the power rule, take the exponent of the argument and bring it forward as a coefficient of the logarithm. The rule can be mathematically represented as:\[\log_b (M^n) = n\cdot\log_b M\]
In practical terms, this means if you have something like \(\log_b (3^4)\), it can be simplified using the power rule to \(4 \cdot \log_b 3\). This simplification is exactly what was done in the given solution.The power rule doesn't just simplify calculations; it also helps in analytical scenarios like solving exponential growth problems or simplifying equations involving exponential functions. By converting complex exponential calculations into more manageable multiplication, the power rule makes dealing with exponents in logs far less daunting.

Logarithmic Approximation

In many cases, it's necessary to estimate logarithmic values rather than calculate them exactly. Sometimes the base of the logarithm isn't common, or the expression is particularly complex. This is where logarithmic approximations come into play.

Logarithmic approximation often involves knowing the approximate values of basic logs thoroughly, which are then used to estimate the value of more complex expressions. For example, if you know that:

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

You can leverage these approximate values to derive other logarithmic expressions.
As demonstrated in the exercise, once you simplify a logarithmic expression using properties like the power rule, you use known approximations to find the estimated values of these simplified expressions. In this case, simplifying \(\log_b 81\) to \(4 \cdot \log_b 3\) and then substituting the known value of \(\log_b 3\) approximates the value effectively. This technique is widely used in scenarios where precision is secondary to practicality, such as estimation challenges, preliminary calculations, and problems requiring quick solution derivation.