Problem 24 - Elementary and Intermediate Algebra

Question

Evaluate expression. \(8^{\log _{8} 10}\)

Step-by-Step Solution

Answer

The value of the expression is 10.

Step 1: Understand the Expression

The expression given is \(8^{\log_{8} 10}\). Here, \(8\) is raised to the power of \(\log_{8} 10\). This expression can be simplified using the properties of logarithms and exponents.

Step 2: Apply Logarithmic and Exponential Properties

According to the properties of logarithms, if \(a^{\log_{a}(b)} = b\), then in this case, since the base of the logarithm and base of the exponent are the same, the expression simplifies directly to \(10\).

Key Concepts

Exponential Expression

An exponential expression is a mathematical expression that has a base raised to a power or exponent. For example, in the expression \(a^b\), \(a\) is the base, and \(b\) is the exponent. This type of expression is fundamental in various areas of mathematics, as it allows us to write repeated multiplication compactly. For instance, \(2^3\) means multiplying 2 by itself three times, equaling 8. Exponential expressions can look complicated at first, particularly when expressions involve variables or complex numbers, but they stick to this simple principle of repeated multiplication.

Understanding exponential expressions is crucial because they offer a way to manage large numbers and simplify calculations. They are often used in science to describe growth processes like population growth or radioactive decay. In our original exercise \(8^{\log_{8} 10}\), the expression utilizes both exponential and logarithmic forms, making it a powerful example of how these concepts are intertwined in mathematical processes.

Simplifying Expressions

Simplifying expressions is the process of making an expression easier to work with by reducing its complexity. This can involve combining like terms, applying mathematical properties or laws, and rewriting the expression in a more manageable form. Simplification helps in making calculations easier and clearer.

In our expression \(8^{\log_{8} 10}\), simplifying involves understanding and applying the relationship between the logarithm and the exponent. Since the base of the logarithm \(8\) matches the base of the exponent \(8\), according to the rule \(a^{\log_a(b)} = b\), the expression directly simplifies to \(10\). This elimination of complexity allows us to see the straightforward result, which is much easier to process than the initial form.

Properties of Exponents

The properties of exponents are rules that describe how to handle expressions involving powers and how exponents can be manipulated. These properties make it easier to work with powers and exponents in mathematical calculations. Here are some important properties to know:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \cdot n}\)
• Zero Exponent: \(a^0 = 1\) for any non-zero \(a\)
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\)

In our expression \(8^{\log_{8} 10}\), one of the properties, which is a combination of logarithm and exponential properties, states that \(a^{\log_a(b)} = b\). This crucial rule simplifies the evaluation of expressions like \(8^{\log_{8} 10}\), illustrating the harmony and connection between exponents and logarithms.