Problem 66 - Precalculus with Calculus Previews

Question

In Problems \(57-68\), use \(\log _{b} 4=0.6021\) and \(\log _{b} 5=\) \(0.6990\) to evaluate the given logarithm. Round your answer to four decimal places. $$ \log _{b} 3.2 $$

Step-by-Step Solution

Answer

\( \log_b 3.2 = 0.5052 \).

Step 1: Express 3.2 in Terms of 4 and 5

To find \( \log_b 3.2 \), first express 3.2 in terms of 4 and 5. Notice that the fraction \( \frac{16}{5} = 3.2 \). So \( 3.2 = \frac{4^2}{5} \).

Step 2: Apply Logarithm Properties

Using the expression \( 3.2 = \frac{4^2}{5} \), apply the properties of logarithms: \( \log_b\left( \frac{4^2}{5} \right) = \log_b(4^2) - \log_b(5) \).

Step 3: Simplify Using Known Logarithms

We know \( \log_b 4 = 0.6021 \) and \( \log_b 5 = 0.6990 \). Use these to find \( \log_b(4^2) = 2\log_b 4 = 2 \times 0.6021 = 1.2042 \).

Step 4: Calculate the Final Logarithmic Value

Substitute the values into the expression: \( \log_b(3.2) = 1.2042 - 0.6990 \). Perform the subtraction to find \( \log_b 3.2 = 0.5052 \).

Key Concepts

Logarithm Properties

Logarithms have some wonderful properties that help simplify complex expressions. Remember, a logarithm is an inverse operation to exponentiation, like subtraction is to addition. Understanding these properties not only makes solving problems straightforward but also enhances your mathematical intuition. One essential property you should remember is the **product property**:

• \( \log_b(M*N) = \log_b(M) + \log_b(N) \)

This property shows how multiplication inside a log can be split into addition outside. You can also flip this property to help with **division**:

• \( \log_b(M/N) = \log_b(M) - \log_b(N) \)

Division within a log becomes subtraction outside. Another important aspect is handling **powers**:

• \( \log_b(N^p) = p \cdot \log_b(N) \)

Exponents can be brought in front of a logarithm, making it a product. These properties allow you to change the form of a logarithm, making it easier to calculate with known values.

Change of Base Formula

Sometimes, you encounter logarithms that aren't in the base you desire. This is when the **change of base formula** comes in handy. It's a clever tool that allows you to convert any log base \( b \) into a different base, commonly 10 or \( e \) (natural logarithm base). The formula is given by:

• \( \log_b(M) = \frac{\log_c(M)}{\log_c(b)} \)

Here, \( c \) can be any new base you choose. By using this formula, you can switch to a base your calculator is most comfortable with. Usually, people convert to base 10, utilizing the common logarithm, which makes it easier if you're using a standard calculator. This formula opens up many possibilities for evaluating logarithms, especially when computational tools are involved that may only work with a specific base.

Evaluating Logarithms

Evaluating logarithms means finding the numeric value of a log expression. It's like asking the question, "To what power should we raise a number (base) to get another number?" In the given problem, we evaluated \( \log_b(3.2) \) by first rephrasing 3.2 in terms of known values. Recognize that solving logs isn't just about arithmetic; it's about clever manipulation using known properties. Often, you re-express numbers to align with known values like our problem does. Here are steps to effectively evaluate logarithms:

• Express the number using known logarithms.
• Apply log properties: product, quotient, and power rules as necessary.
• Simplify using given log values.

By combining these strategies, like expressing 3.2 as \( \frac{4^2}{5} \), calculating becomes straightforward. Using known log values 0.6021 and 0.6990, we calculate \( \log_b(3.2) \) easily. Understanding how to manipulate and evaluate logs provides not only a means to solve problems but also a deeper appreciation of mathematical relationships.