Question

\log _{a} a^{r}= __________

Short answer

\log_{a} a^{r} = r

Step-by-step solution

Identify the Equation

The given equation is \(\log_{a} a^{r} = ?\). It involves a logarithm with a base of \(a\).

Apply the Power Rule of Logarithms

According to the power rule of logarithms, \(\log_{b} b^{k} = k\), where \(b\) is the base and \(k\) is the exponent. Here, \(b = a\) and \(k = r\).

Simplify the Expression

Using the power rule, we can simplify \(\log_{a} a^{r}\) to just \(r\). Thus, \(\log_{a} a^{r} = r\).

Key concepts

Logarithms

A logarithm is a mathematical operation that helps you find the exponent needed to obtain a given number. When you see \(\text{log}_{b}(x)\), it means 'to which power should \ b\ be raised to get \ x\?'.
Logarithms have various properties that make them exceptionally useful in different areas of math and science. For example, in our exercise, we are working with the power rule of logarithms.

Some important properties of logarithms include:

• Product Property: \( \text{log}_{a}(xy) = \text{log}_{a}(x) + \text{log}_{a}(y) \)
• Quotient Property: \(\text{log}_{a}(x/y) = \text{log}_{a}(x) - \text{log}_{a}(y) \)
• Power Rule: \(\text{log}_{a}(x^r) = r\text{log}_{a}(x) \)

Understanding these properties will give you tools to manipulate and simplify logarithmic expressions with ease. In our specific case, we applied the power rule to find that the expression simplifies to \ r \.

Exponentiation

Exponentiation is the process of raising a number to a power. This operation involves two key components: the base and the exponent.
For example, in the expression \(a^{r}\), \ a\ is the base and \ r\text{ is the exponent.}

The exponent tells you how many times to multiply the base by itself. For instance, \(2^3\) means \(2 \times 2 \times 2 = 8\). Exponentiation is crucial in various fields, from science to finance, because it describes how quantities grow over time, such as compound interest or population growth.

Key points to remember about exponentiation:

• The base can be any real number.
• The exponent is often a positive integer, but it can also be zero, a negative number, or even a fraction.

By understanding exponentiation, you can more easily grasp why \( \text{log}_{a} a^{r} = r\). Essentially, logarithms and exponentiation are inverse operations.

Simplifying Expressions

Simplifying expressions means making them easier to work with. It involves using properties of arithmetic and algebra to reduce complexity. For logarithmic expressions, simplification often involves applying rules like the power rule, product rule, and quotient rule.

Let's look at our exercise: \( \text{log}_{a} a^{r} \). Applying the power rule of logarithms, \( \text{log}_{b} b^{k} = k \), helps us simplify this naturally. This rule states that if you take a logarithm of a number that is already expressed as that logarithm's base raised to a power, it simplifies to that exponent.

Simplification steps:

• Recognize the form of the expression: \( \text{log}_{a} a^{r} \).
• Identify relevant properties or rules: Power Rule.
• Apply the rule: \( \text{log}_{a} a^{r} = r \).

This makes the expression \( r \). Simplifying expressions saves time and computational effort, making it easier to solve complex problems. It also helps in forming a deeper understanding of the relationships between mathematical concepts.