Problem 77 - Precalculus with Calculus Previews

Question

In Problems 77 and 78 , find the numerical value of the given logarithm. $$ \log _{\pi} 4 $$

Step-by-Step Solution

Answer

\( \log_{\pi} 4 \approx 1.211 \)

Step 1: Identify the base and argument

The problem given is \( \log_{\pi} 4 \). Here, the base is \( \pi \) and the argument is \( 4 \). The task is to find a numerical value \( x \) such that \( \pi^x = 4 \).

Step 2: Use the change of base formula

The change of base formula for logarithms is given by: \[ \log_b a = \frac{\log_k a}{\log_k b} \] where \( k \) is any positive number (commonly taken as 10 or \( e \)). Let's use natural logarithms (base \( e \), denoted as \( \ln \)): \( \log_{\pi} 4 = \frac{\ln 4}{\ln \pi} \).

Step 3: Calculate logarithmic values

Using a calculator, find the natural logarithms \( \ln 4 \) and \( \ln \pi \). These approximate values are: \( \ln 4 \approx 1.3863 \) and \( \ln \pi \approx 1.1447 \).

Step 4: Compute the division

Substitute the values into the expression from the change of base formula: \( \frac{\ln 4}{\ln \pi} = \frac{1.3863}{1.1447} \).

Step 5: Find the numerical result

Calculate the division to find the approximate value of the logarithm: \( \frac{1.3863}{1.1447} \approx 1.211 \). Thus, \( \log_{\pi} 4 \approx 1.211 \).

Key Concepts

Change of Base Formula

The Change of Base Formula is a handy mathematical tool that lets us rewrite logarithms in terms of bases that are easier to calculate, like base 10 or base \( e \) (natural logarithms). This formula is given by:

• \( \log_b a = \frac{\log_k a}{\log_k b} \)

Here, \( b \) is the base of the original logarithm, \( a \) is the argument, and \( k \) is the base to which we want to change, often 10 or \( e \).
Using this formula helps us evaluate logarithms that aren't straightforward to calculate directly with a calculator.
For example, in the original problem, we used this formula to change from base \( \pi \) to base \( e \), giving us \( \log_{\pi} 4 = \frac{\ln 4}{\ln \pi} \). By using natural logarithms, we can make use of calculator functions that are readily accessible, allowing for more efficient calculations of otherwise complex expressions.

Natural Logarithms

Natural logarithms, denoted as \( \ln \), are logarithms with base \( e \), where \( e \) is approximately equal to 2.71828. This constant is widely used in mathematics because of its natural properties and the fact that it arises in many real-world scenarios.
Natural logarithms have some unique characteristics:

• The derivative of \( \ln x \) is \( \frac{1}{x} \), which has useful properties in calculus.
• \( \ln e = 1 \) because \( e^1 = e \).
• They simplify the handling of exponential growth and decay models.

In the context of the original problem, natural logarithms are used in the change of base formula to compute \( \log_{\pi} 4 \). By using \( \ln 4 \) and \( \ln \pi \), we simplify the calculation process since natural logarithm values are easy to obtain with calculators.

Exponential Equations

Exponential equations are those where the variable is in an exponent. A basic form of an exponential equation is \( b^x = a \). Solving these types of equations often involves using logarithms to "bring down" the exponent.
This is because logarithms are the inverse operations of exponentials, allowing us to express the exponent in terms of more manageable algebraic expressions:

• For example, solving \( \pi^x = 4 \) by taking the logarithm of both sides lets us find \( x \).

In problems such as the given one, understanding and manipulating exponential relationships through logarithms enable us to solve for unknowns even when the base is a more unusual number like \( \pi \). This underscores the importance of understanding both exponential and logarithmic functions deeply.