Problem 77 - Algebra and Trigonometry

Question

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{\pi} 63 $$

Step-by-Step Solution

Answer

The final answer is approximately 3.6205.

Step 1: Understand the change of base formula

The change of base formula is given by \[ \log_b a = \frac{\log_c a}{\log_c b} \] where any positive base c except 1 can be used. This formula will be needed to continue.

Step 2: Substitute into the formula

Remembering the task to find \(\log_\pi 63\), now \(\pi\) will be set as the base (b) and 63 as the argument of the logarithm (a). To simplify calculations, the natural logarithm will be chosen (c = e). So, substitute these values into the change of base formula, given in step 1. This substitution process gives: \[ \log_\pi 63 = \frac{\log_e 63}{\log_e \pi} \]

Step 3: Calculate the values using a calculator

Now, using a calculator or computational software, evaluate the natural logarithm of the numbers 63 and \(\pi\) (the number 63 in the numerator and \(\pi\) in the denominator). After calculating, you should get the following: \[ \log_\pi 63 \approx \frac{4.143135}{1.1447299} \]

Step 4: Final Calculation

Continuing the calculation, divide the values obtained in the previous step to get the final answer. That gives: \[ \log_\pi 63 \approx 3.6205 \]

Key Concepts

Change of Base Formula

The change of base formula is a handy tool when dealing with logarithms, especially with less common bases like \( \pi \). It enables us to express a logarithm with one base in terms of logarithms with another base. This is how it works:

• The formula is \( \log_b a = \frac{\log_c a}{\log_c b} \).
• Here, \( b \) is the base of your original logarithm, \( a \) is the argument, and \( c \) is the new base you choose, usually 10 (common logarithm) or \( e \) (natural logarithm).

This way, the calculation becomes manageable with common logarithms and natural logarithms, which are supported by most calculators. For our problem, evaluating \( \log_\pi 63 \), the change of base formula transforms it to an easier computation using natural logs.

Natural Logarithms

Natural logarithms (often written as \( \ln \)) use the number \( e \) as their base. The number \( e \) (approximately 2.718) is a fascinating constant that appears in various areas of mathematics, especially calculus.

• Natural logarithms are written as \( \ln(x) \) and mean the same as \( \log_e(x) \).
• They simplify the calculation process because calculators are designed to compute them quickly.

In our exercise, we're using natural logarithms to take advantage of the change of base formula:

• We substitute \( 63 \) and \( \pi \) into the formula, \( \frac{\ln 63}{\ln \pi} \).

This approach is why you often see \( e \) being used with the change of base formula.

Calculating Logarithms with a Calculator

Calculators are essential tools when working with logarithms, especially when decimals or irrational numbers like \( \pi \) are involved. Here’s how you can efficiently evaluate these logarithms:First, ensure your calculator has a function for natural logarithms (\( \ln \)). Most scientific calculators have this feature.

• To find \( \ln 63 \), enter the number 63 and press the \( \ln \) button.
• Do the same for \( \pi \) by either entering the \( \pi \) symbol directly (available on many calculators) or typing 3.14159 and pressing the \( \ln \) button.

So for the problem \( \log_\pi 63 \), plug these values into our adjusted expression \( \frac{\ln 63}{\ln \pi} \). After dividing, your calculator should give you the final answer, accurate to four decimal places as \( 3.6205 \). Confirming these calculations with your device ensures accuracy, especially where precision is key.