Problem 23 - Elementary and Intermediate Algebra

Question

Evaluate expression. \(5^{\log _{5} 10}\)

Step-by-Step Solution

Answer

The expression evaluates to 10.

Step 1: Understanding the Expression

The expression presented is \(5^{\log _{5} 10}\). This is an exponential expression where the base of the exponent is the same as the base of the logarithm. This suggests a specific logarithmic property can be applied.

Step 2: Applying the Logarithm Power Rule

Recall the logarithm power rule: If \(b^{\log_b (x)} = x\), then the expression simplifies directly to \(x\), because the power and the logarithmic function cancel each other out due to their inverse nature.

Step 3: Simplifying the Expression

In our expression, the base of the logarithm (\(5\)) is the same as the base of the exponent. Therefore, using the property from the previous step, we simplify \(5^{\log _{5} 10}\) to just \(10\).

Key Concepts

Exponential Functions

Exponential functions are a foundational concept in mathematics. They involve expressions where a variable serves as the exponent, typically written as \(b^x\). Here, \(b\) is the base and \(x\) is the exponent.
Understanding this structure is crucial because exponential functions demonstrate how quantities grow rapidly and appear in various scientific fields, including finance and population dynamics.
Key characteristics of exponential functions include:

• Rapid Increase: As the value of \(x\) increases, \(b^x\) grows quickly if \(b > 1\), and decreases if \(0 < b < 1\).
• Constant Base: The base \(b\) remains unchanged throughout the function.
• Exponential Growth and Decay: These terms describe how quantities increase or decrease exponentially.

Recognizing exponential functions can help identify patterns within complex systems. In the given exercise, \(5^{\log _{5} 10}\) is a specific form of an exponential function applied with logarithmic expressions.

Logarithmic Properties

Logarithms are the inverse operations of exponentiation, allowing us to solve equations involving exponential expressions. They help translate multiplicative processes into additive ones, making complex calculations simpler.
There are several key properties of logarithms that simplify expressions:

• Product Rule: \(\log_b(MN) = \log_b(M) + \log_b(N)\)
• Quotient Rule: \(\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)\)
• Power Rule: \(\log_b(M^n) = n \cdot \log_b(M)\)

These properties allow for manipulating and simplifying logarithmic expressions, especially when combined with exponential functions. In the original exercise, the understanding of these properties, particularly the relationship between the base of the logarithm and the base of the exponential expression, helps to simplify \(5^{\log _{5} 10}\).

Power Rule for Logarithms

The power rule for logarithms is especially useful when dealing with expressions where the logarithm appears as an exponent. It states that \(b^{\log_b(x)} = x\). This is because the logarithm and the exponential function are inverses of each other.
Here's how the power rule simplifies expressions:

• Cancellation Effect: The logarithmic and exponential functions cancel each other out, as they are inverse functions.
• Direct Simplification: In expressions such as \(b^{\log_b(x)}\), the result is directly \(x\), as seen in our original exercise.

Understanding and applying the power rule makes it much easier to handle complex expressions. In the exercise, \(5^{\log _{5} 10}\) directly becomes \(10\) using this power rule, showing its simplicity and effectiveness.