Problem 27 - Precalculus with Limits

Question

Use the properties of logarithms to simplify the expression. \(\log _{\pi} \pi\)

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Acknowledge the properties of log

Recognize that this expression takes the form of the log identity \(\log_{b}b = 1\), where \(b\) is any base. Here, the base \(b\) is \(\pi\). This is the property of logarithm which says that the logarithm of any number to its own base is always equal to 1.

Step 2: Apply the log property

Apply the identity property of logarithm, yielding us the simplified value of 1.

Key Concepts

Logarithm Identity

A logarithm identity is a fundamental property that simplifies logarithmic expressions. The identity we are discussing here is \( \log_{b} b = 1 \). This means when you take the logarithm of a number using the same number as the base, the result is always 1.

Here's why this makes sense:

• The logarithm tells us the power or exponent needed to get one number from another.
• So, \( \log_{b} b \) asks: "What power do we raise \( b \) to, in order to get \( b \)?"
• The answer is 1 because \( b^1 = b \).

Applying this identity makes complex calculations much more straightforward. Always remember that this identity is handy when dealing with logarithms where the base and the number inside the log are the same.

Logarithmic Base

Understanding the base of a logarithm is crucial to simplify and solve logarithmic expressions. The base is the number we are using as a reference when determining the power to achieve another number.

Think of it like this:

• In the expression \( \log_{b} x \), \( b \) is the base.
• The base must always be positive and not equal to 1.
• In a base \( b \), we find the power to which \( b \) must be raised to obtain a number \( x \).

For example, in \( \log_{\pi} \pi \), the base is \( \pi \). Since the base and the number are the same, it simplifies using the logarithm identity \( \log_{b} b = 1 \). Understanding the base helps break down the expression efficiently.

Simplifying Logarithms

Simplifying logarithms involves using known properties to make expressions easier to understand and work with. The main goal of simplification is to transform complex logarithmic expressions into simpler forms.

Here are steps to simplify logarithms effectively:

• Use logarithm identities like \( \log_{b} b = 1 \) to instantly recognize when expressions equal to 1.
• Combine or break apart logarithms using rules like the product, quotient, and power rules.
• Simplify expressions by recognizing patterns or using mathematical operations.
• Check if the base and the argument (the number inside the log) are the same.

In our example, \( \log_{\pi} \pi \), the simplification is immediate because of the identity. Always look for opportunities to apply these rules to save time and avoid lengthy computations.