Problem 62 - Precalculus Mathematics for Calculus

Question

Use the Change of Base Formula and a calculator to evaluate the logarithm, rounded to six decimal places. Use either natural or common logarithms. $$\log _{12} 2.5$$

Step-by-Step Solution

Answer

\( \log_{12} 2.5 \approx 0.368057 \)

Step 1: Identify the Base and Argument

The given logarithm is \( \log_{12} 2.5 \). Here, the base is 12, and the argument is 2.5.

Step 2: Apply the Change of Base Formula

The Change of Base Formula is: \( \log_b a = \frac{\log_c a}{\log_c b} \). We can use either common logarithms (base 10) or natural logarithms (base \( e \)).

Step 3: Choose the Type of Logarithm

Let's use common logarithms (base 10) to apply the formula. Thus, we rewrite: \( \log_{12} 2.5 = \frac{\log 2.5}{\log 12} \).

Step 4: Calculate Both Logarithms

Calculate \( \log_{10} 2.5 \) and \( \log_{10} 12 \) using a calculator. We find \( \log_{10} 2.5 \approx 0.397940 \) and \( \log_{10} 12 \approx 1.079181 \).

Step 5: Compute the Quotient

Divide the two calculated logarithms: \( \frac{0.397940}{1.079181} \approx 0.368057 \).

Step 6: Round the Result

Round the answer obtained to six decimal places. Therefore, \( \log_{12} 2.5 \approx 0.368057 \).

Key Concepts

Common Logarithms

Common logarithms are logarithms with base 10. This is symbolized as \( \log_{10} \) or often simply written as \( \log \) when the base is understood to be 10.
Common logarithms are widely used because the decimal system is base 10, making them convenient for working with numbers typically encountered in daily life.

When you see \( \log 100 \), it means "What power do I raise 10 to in order to get 100?" The answer is 2, since \( 10^2 = 100 \). This operation simplifies calculations, especially when using a common calculator or certain mathematical software.

• Common logarithms are handy in science and engineering because they're aligned with the decimal system.
• They help express large numbers in a more manageable form, which is useful in fields like physics and chemistry.

In the solution to the original exercise, the common logarithm was used to convert \( \log_{12} 2.5 \) through the Change of Base Formula, resulting in the quotient of two base 10 logarithms.

Natural Logarithms

Natural logarithms have base \( e \), where \( e \approx 2.71828 \). The notation for them is \( \ln \). Natural logarithms are often more prevalent in advanced mathematics and calculus, particularly when dealing with continuous growth or decay.

They offer a straightforward way to work with exponential growth processes because the base \( e \) occurs naturally in mathematics, particularly in compound interest calculations and various natural phenomena.

• Natural logarithms show up in problems involving growth and decay, like population growth, radioactive decay, and more.
• They are used in fields ranging from physics to finance.

While the exercise used common logarithms through the Change of Base Formula, you could also have employed natural logarithms to achieve the same result, as \( \ln \) functions similarly when applying the Change of Base Formula.

Base 10 Logarithms

Base 10 logarithms, also known as common logarithms, are essential for converting between different log bases.
The Change of Base Formula is particularly helpful for calculations involving non-standard log bases. When a logarithm isn't base 10 or base \( e \), this formula lets you convert it using common or natural logarithms.

In mathematical notation, the Change of Base Formula is \( \log_b a = \frac{\log_c a}{\log_c b} \), allowing the transformation of a log with any base into a quotient of logarithms that can be computed with a calculator.

• This is useful for solving logarithmic equations where the base is not immediately accessible by standard calculators.
• Common scenarios include determining unknowns in exponential functions.

In the original exercise, this formula was used effectively to determine \( \log_{12} 2.5 \), demonstrating the utility of base 10 logarithms in simplifying complex calculations.