Problem 23 - Intermediate Algebra

Question

In this Study Set, assume that all variables represent positive numbers and \(b \neq 1\). Evaluate each expression. See Example \(1 .\) $$ 5^{\log _{5} 10} $$

Step-by-Step Solution

Answer

The expression evaluates to 10.

Step 1: Review Exponential and Logarithm Properties

Recall that for any positive number \(a\) where \(aeq1\), the expression \(a^{\log_{a}b} = b\) because the logarithm represents the exponent needed for \(a\) to become \(b\).

Step 2: Apply the Property to the Expression

To evaluate \(5^{\log_{5}10}\), use the previously mentioned property of exponents and logarithms. Given the expression, identify \(a = 5\) and \(b = 10\). Then \(5^{\log_{5}10} = 10\).

Step 3: Verify the Solution

Since \(5^{\log_{5}10} = 10\) follows directly from the property of exponents and logarithms, check if substituting these values satisfy the condition \(a^{\log_{a}b} = b\) by making sure the bases and logarithms match, which they do in this case.

Key Concepts

Properties of Exponents

Exponents are a way to express repeated multiplication of a number by itself. The base is the number being multiplied, and the exponent tells how many times to multiply the base by itself. The properties of exponents are handy for simplifying and solving expressions involving exponents.

Here are some of the fundamental properties:

• Product of Powers: When you multiply two expressions with the same base, you add the exponents: \(a^m imes a^n = a^{m+n}\).
• Power of a Power: When raising an exponent to another exponent, multiply the exponents: \((a^m)^n = a^{m imes n}\).
• Power of a Product: When raising a product to an exponent, apply the exponent to each factor: \((ab)^n = a^n b^n\).
• Quotient of Powers: Divide two expressions with the same base by subtracting the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).

By understanding and applying these properties, you can simplify complex expressions and solve equations more effectively. The particular property used in our exercise is that when a base is raised to the logarithm of a number with the same base, like \(a^{\log_a b}\), the result is simply \(b\). This is known as the inverse property of logarithms and exponents.

Properties of Logarithms

Logarithms are the inverse operations of exponents. They answer the question, "To what power must a base be raised, to produce a given number?" Understanding their properties is key to manipulating and simplifying logarithmic expressions.

Here are some important logarithmic properties:

• Product Property: The logarithm of a product equals the sum of the logarithms: \(\log_b(mn) = \log_b m + \log_b n\).
• Quotient Property: The logarithm of a quotient equals the difference of the logarithms: \(\log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n\).
• Power Property: The logarithm of a power means multiplying the exponent by the logarithm: \(\log_b(m^n) = n \cdot \log_b m\).
• Change of Base Formula: You can change the base of a logarithm using: \(\log_b a = \frac{\log_k a}{\log_k b}\), with any base \(k\).

Utilizing these properties allows you to transform and simplify logarithmic expressions efficiently. In the context of our exercise, understanding the inverse relationship between logarithms and exponents is crucial. When you see \(5^{\log_5 10}\), it immediately simplifies to 10, thanks to the properties described above.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions involves using the properties of logarithms and their relationship with exponents. Our exercise used the property \(a^{\log_a b} = b\) to evaluate expressions like \(5^{\log_5 10}\). This is an application of the inverse nature of exponents and logarithms.

To evaluate any logarithmic expression:

• Identify the Bases: Ensure that the bases in your logarithm and exponent match as in \(a\) and \(\log_a\).
• Apply Properties: Use relevant properties of exponents and logarithms to simplify the expression. In this case, the exponent-logarithm inverse property is most useful.
• Double-Check Bases: Verify that the expression correctly follows our key property by matching \(a^{\log_a b} = b\) with your numbers. This confirms correct simplification.

The ability to shift flexibly between exponential and logarithmic forms simplifies solving logarithmic expressions. By mastering these key ideas, you can deal with more advanced mathematical problems with confidence.