Problem 41 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression without using a calculator.\(\log _{a} a^{5}\)

Step-by-Step Solution

Answer

The result of this logarithmic expression, \(\log _{a} a^{5}\), is \(5\).

Step 1: Identify Base and Exponent

In the given expression \(\log _{a} a^{5}\), the base is \(a\) and the exponent is \(5\).

Step 2: Apply Properties of Logarithms

According to the rule \(\log _{a} a^{n}=n\), if the base of the logarithm and the number are the same, then the result is simply the exponent. Therefore, \(\log _{a} a^{5}\) equals to \(5\).

Key Concepts

Properties of Logarithms

Logarithms have several important properties that help simplify expressions and solve equations efficiently. Understanding these properties is key to mastering logarithmic expressions. One of the fundamental properties is the identity property, which states that if you have a logarithm with a base \(b\) and you are taking the logarithm of \(b^n\), the result is simply \(n\). Mathematically, it is represented as:

• \(\log_b (b^n) = n\)

This property is crucial when bases in logarithmic and exponential forms match, allowing for direct simplification. It essentially works by reversing the process of exponentiation, stripping away the base and leaving the exponent.
Another useful property is the product property, which allows us to add logarithms when multiplying numbers:

• \(\log_b (xy) = \log_b (x) + \log_b (y)\)

These fundamental properties can be greatly helpful when breaking down complex logarithmic expressions into more manageable parts.

Exponentiation

Exponentiation refers to the process of raising a number to the power of another number. This is a core concept in mathematics, where the number being multiplied, known as the base, is multiplied by itself a number of times indicated by the exponent. For example, in the expression \(a^5\), the base is \(a\) and the exponent is 5.
Exponentiation is shown as:

• \(x^n\) where \(x\) is the base and \(n\) is the exponent

When we talk about logarithms, they are essentially the inverse operation of exponentiation. Logarithms help us "undo" exponentiation by finding the exponent.
Understanding exponentiation is critical when dealing with logarithms because many logarithmic properties derive from the principles of exponents, such as simplifying logarithms of products, quotients, and powers. Learning to "recognize" expressions like \(a^5\) allows us to use logarithmic properties effectively.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions involves applying properties of logarithms to simplify and find the value of a logarithmic statement. Logarithms tell us what the power is that a number must be raised to, in order to get another number. When evaluating logarithmic expressions, it often involves:

• Recognizing the base and its corresponding exponent
• Applying the relevant logarithmic properties to simplify the expression
• Calculating the result, if needed, without a calculator

For instance, in the expression \(\log_{a} a^{5}\), the task is to determine "to what power must \(a\) be raised to get \(a^5\)?" The answer is simply 5, because of the identity property which states \(\log_b(b^n) = n\).
Evaluating requires practice in recognizing and applying these rules. The fewer steps one needs to compute the result, the easier it becomes to solve real world logarithmic problems. Always look out for opportunities to leverage properties like the identity property to quickly find solutions.