Problem 56 - Precalculus

Question

Evaluate the logarithms using the change-of-base formula. Round to four decimal places. $$\log _{\pi} 2.7$$

Step-by-Step Solution

Answer

\(\log_{\pi} 2.7 \approx 0.8677\)

Step 1: Identify the change of base formula

The change of base formula is used to evaluate logarithms with bases other than 10 or e. It is given by: \[ \log_b a = \frac{\log_c a}{\log_c b}\] where \(c\) is a new base, commonly 10 or \(e\) for natural logarithms.

Step 2: Apply the formula to the given problem

We need to evaluate \(\log_{\pi} 2.7\). By applying the change of base formula, we get: \[\log_{\pi} 2.7 = \frac{\log 2.7}{\log \pi}\] We can use either common logarithm (base 10) or natural logarithm (base \(e\)).

Step 3: Calculate the numerators and denominators

Calculate the common logarithm of 2.7: \[\log 2.7 \approx 0.4314\]Next, calculate the common logarithm of \(\pi\): \[\log \pi \approx 0.4971 \]

Step 4: Perform the division

Substitute the values into the formula: \[\log_{\pi} 2.7 = \frac{0.4314}{0.4971}\]Perform the division to get:\[\log_{\pi} 2.7 \approx 0.8677\]

Step 5: Round the answer

Round the final result to four decimal places. The result is already rounded, so we have: \[\log_{\pi} 2.7 \approx 0.8677\]

Key Concepts

Understanding Logarithms

Logarithms are incredibly useful in mathematics, especially when it comes to dealing with exponential functions. Essentially, a logarithm answers the question "how many times does one need to multiply a base by itself to get a certain number?" For instance, in the expression \( \log_b a \), we are seeking the power that the base \( b \) must be raised to, in order to yield \( a \). The base \( b \) is a crucial component, as it helps define the logarithm's behavior. Common bases include 10, \( e \), and any positive number greater than zero. This makes logarithms a very flexible tool in both theoretical and applied contexts.

Here's a breakdown of key points about logarithms:

• The base of a logarithm must be greater than zero and cannot be 1.
• Logarithms are the inverse operation of exponentiation.
• Understanding base changes is critical for solving more complex logarithmic equations.

When having complex bases or numbers, we often apply the change of base formula. It allows us to transform a logarithm into a different base that might be easier to handle or calculate, such as 10 or the natural base \( e \).

Exploring the Natural Log

The natural logarithm is denoted by \( \ln \) and is based on the number \( e \), approximately equal to 2.71828. This form of logarithm is vital across various fields, thanks to its natural occurrence in processes involving growth and decay.

Why do we use the natural log?

• It simplifies expressions that deal with exponential growth or decay.
• Often appears naturally in problems involving calculus and differential equations.
• Its derivative is straightforward, making it a favorite in calculus.

The natural log can invert exponential functions with base \( e \). It's worth noting that using the natural log in the change-of-base formula is as valid as using the common log. With \( \,\ln \), you can simplify expressions while retaining a powerful tool for different calculations.

Importance of the Common Log

The common logarithm is denoted by \( \log \) and typically uses base 10. It's one of the most frequent types of logarithms you'll encounter. The common log's familiarity makes it particularly useful in everyday calculations and scientific computations.

Some highlights about the common log include:

• Base 10 aligns perfectly with the decimal number system, making arithmetic calculations more intuitive.
• Usually used in scientific notation to express large or small numbers in a compact form.
• Essential in fields like chemistry, physics, and engineering, where scale handling is crucial.

In the change-of-base formula, the common log helps evaluate logarithms of unconventional bases. Using the formula \( \log_b a = \frac{\log_c a}{\log_c b} \), swapping in base 10 as \( c \) provides a straightforward method for computation. This versatility is a significant advantage when dealing with more complex logarithmic equations.