Problem 34 - Precalculus

Question

In Exercises 34 - 39 , use the appropriate change of base formula to approximate the logarithm. $$ \log _{3}(12) $$

Step-by-Step Solution

Answer

\( \log_{3}(12) \approx 2.26186 \) using base 10 logarithms.

Step 1: Understand the Problem

We need to find an approximate value for the logarithm \( \log_{3}(12) \). Since this isn't a simple integer, we will use the change of base formula to compute it.

Step 2: Apply Change of Base Formula

The change of base formula states that \( \log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)} \), where \( c \) is a new base you choose. Commonly, \( c = 10 \) (common log) or \( c = e \) (natural log). We'll use base 10 for this problem.

Step 3: Write the Expression Using Base 10

Using the change of base formula with base 10, we have:\[\log_{3}(12) = \frac{\log_{10}(12)}{\log_{10}(3)}\]

Step 4: Calculate Logarithms Separately

Find the values of \( \log_{10}(12) \) and \( \log_{10}(3) \) using a calculator or log table:- \( \log_{10}(12) \approx 1.07918 \) - \( \log_{10}(3) \approx 0.47712 \)

Step 5: Divide the Results

Now, divide the values obtained: \[\frac{1.07918}{0.47712} \approx 2.26186\]

Step 6: State the Final Approximation

Thus, \( \log_{3}(12) \approx 2.26186 \) to five decimal places.

Key Concepts

Understanding Logarithms

Logarithms are essentially the inverse operations of exponentiation. If you're familiar with powers and exponents, understanding logarithms will feel like working backwards. They're used to determine what power you need to raise a base to in order to obtain a specific number. For instance, if we say \( \, b^x = a \, \), then \( \, ext{log}_b(a) = x \, \). Here's a breakdown:

• The base \( b \) is the number you multiply by itself.
• The exponent \( x \) is the number of times you multiply the base.
• \( a \) is the result after raising the base to the exponent.

In our context of finding \( \log_{3}(12) \), we're trying to determine what power we must raise 3 to obtain 12. However, 12 isn't a power of 3 that results in a whole number exponent, hence the need for approximation.

The Process of Base Conversion

Base conversion in logarithms refers to changing the base of a logarithm to another base that might be easier to work with. This is helpful because calculating logarithms with bases other than 10 or \( e \) directly is often impractical. The change of base formula allows you to convert a logarithm to a new base of your choice.

• The formula for base conversion is \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \), where \( c \) is your chosen new base, usually 10 or \( e \) for convenience.
• This formula works because logarithms are ratios of the exponents needed to reach a certain number from a given base. Thus, changing the base doesn't change the ratio of these exponents.

By converting \( \log_{3}(12) \) to a base of 10, we make the calculation possible by using calculators or tables designed for these common bases.

Logarithmic Calculations in Practice

Now let's apply our understanding to the practical calculation of logarithms using the change of base formula. When you use a base-10 calculator or tools, the process becomes straightforward. Here's a step-by-step method:

• First, calculate \( \log_{10}(12) \) which approximates to 1.07918. This is finding how many times 10 has to be raised to reach 12.
• Next, calculate \( \log_{10}(3) \) which approximates to 0.47712. It tells us how many times 10 must be raised to equal 3.
• Finally, divide these two results: \( \frac{1.07918}{0.47712} \). This gives you approximately 2.26186, which means 3 raised to 2.26186 equals about 12.

With the change of base formula, complex logarithmic calculations become manageable, making it crucial for advancing in mathematics and related fields.