Problem 102 - Elementary and Intermediate Algebra

Question

Use the change-of-base formula to find logarithm to four decimal places. \(\log _{\pi} e\)

Step-by-Step Solution

Answer

\(\log_\pi e \approx 0.8731\)

Step 1: Understand the Change-of-Base Formula

The change-of-base formula allows you to evaluate a logarithm with any base by expressing it in terms of logarithms with a different base, usually base 10 or base e (the natural logarithm). The formula is: \[\log_b a = \frac{\log_k a}{\log_k b}\]where \(b\) is the base of the original logarithm, \(a\) is the value whose logarithm you want to find, and \(k\) is the new base you choose (commonly 10 or \(e\)).

Step 2: Apply the Change-of-Base Formula

In this exercise, we want to find \(\log_\pi e\). Using the change-of-base formula, we choose base \(e\) (natural logarithm) for convenience:\[\log_\pi e = \frac{\ln e}{\ln \pi}\]We apply the natural logarithm \(\ln\) to both \(e\) and \(\pi\).

Step 3: Evaluate the Natural Logarithms

Calculate both \(\ln e\) and \(\ln \pi\): - \(\ln e = 1\), because the natural logarithm of Euler's number \(e\) is 1 by definition.- Use a calculator to find \(\ln \pi\), which is approximately 1.1447 to four decimal places.

Step 4: Calculate \(\log_\pi e\)

Substitute the evaluated logarithm values into the change-of-base formula: \[\log_\pi e = \frac{1}{1.1447}\]Calculate the result to find \(\log_\pi e \approx 0.8731\) when rounded to four decimal places.

Key Concepts

Logarithms

Logarithms are mathematical tools used to solve for powers. They are the inverse operation of exponentiation, meaning that they help you find the exponent that a base number is raised to in order to reach another number. For example, if you know that \(b^c = a\) (where \(b\) is the base, \(c\) is the exponent, and \(a\) is the result), then \(\log_b a = c\) tells you what \(c\) is.

• Base: The number that is raised to a power.
• Exponent: The power to which the base number is raised.
• Logarithm: The exponent itself.

While many students initially see logarithms as challenging, they're quite similar to operations you already know, like division being the opposite of multiplication. Understanding the concept of a base, or the fixed number, is essential for working with logs. This is because the base defines the rate and the system in which the logarithms work, much like how the base 10 works for our everyday number system.

Natural Logarithm

Natural logarithms are a specific kind of logarithm with a base of the special number \(e\). The number \(e\) is an irrational and transcendental number approximately equal to 2.71828.
They are denoted as \(\ln\) and are particularly compelling for calculations involving continuous growth processes like compound interest or population growth.

• Base \(e\): A constant that appears widely in mathematics, especially in calculus.
• Symbol \(\ln\): Used to denote natural logarithms.

Using natural logarithms simplifies calculations in mathematics, making complex expressions more manageable. They often appear in equations that model real-world phenomena, where growth or decay is continuous. Moreover, the property \(\ln(e) = 1\) is particularly useful, as it becomes a simplification in many calculations. Thus, understanding \(\ln\) helps tackle problems in exponential growth and decay, especially those where calculating by other means would be unwieldy.

Base Conversion

Base conversion in terms of logarithms involves changing from one logarithmic base to another, often for ease of calculation. This is particularly useful because calculators typically feature base 10 (common logarithms) and natural logarithms (base \(e\)), but not for other bases like \(\pi\).
The change-of-base formula is:
\[\log_b a = \frac{\log_k a}{\log_k b}\]Here, \(b\) is the old base, \(a\) is the number you're finding the logarithm of, and \(k\) is the new base. This formula permits the calculation of any logarithm using your desired base (commonly 10 or \(e\)).

• Old Base (\(b\)): The original base of your logarithm.
• Value (\(a\)): The number you need the logarithm for.
• New Base (\(k\)): The base to which you're converting.

For instance, finding \(\log_\pi e\) using \(\ln\) involves dividing \(\ln e\) by \(\ln \pi\), which is why understanding base conversion is so crucial. By simply using a change-of-base formula, we can convert any complicated logarithm into a simpler form to solve efficiently using standard tools like calculators.