Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

Simplify the given expression. $$ 10^{2 \log _{10} 5} $$

Step-by-Step Solution

Answer

The simplified expression is 25.

Step 1: Understand the Expression

The expression given is \(10^{2 \log_{10} 5}\). This is an exponential expression in which the base is raised to a specific power.

Step 2: Use the Property of Logarithms

Recall that \(a^{\log_b c} = c^{\log_b a}\). Therefore, knowing the property \(a^{b \cdot \log_c d} = (d)^{\log_c a^b}\), we can rewrite the expression as \((5^{\log_{10} 10^2})\).

Step 3: Simplify Using Logarithm Property

Since \(\log_{10} 10 = 1\), \(\log_{10} 10^2 = 2 \cdot \log_{10} 10 = 2 \cdot 1 = 2\). Now substitute back in: \(5^{2}\).

Step 4: Simplify the Final Expression

Finally, compute \(5^2\). This gives us \(5 \times 5 = 25\). Thus, the simplified form of the expression is 25.

Key Concepts

Logarithmic Property

Logarithmic properties are fundamental tools in mathematics that simplify complex exponential expressions. They serve as a bridge between exponential and polynomial forms, making calculations and simplifications more intuitive. A key property is that the logarithm of a power allows us to manipulate expressions such as \(a^{b \cdot \log_c d}\). This property tells us that if we have a logarithm exponentiated by a power, it can be rewritten as another exponentiation. In our original expression, we relied on this property:

• First, notice that \(10^{2 \log_{10} 5}\) can be interpreted as \((5^{\log_{10} 10^2})\).
• The transformation uses the logarithm power rule, \(a^{b \cdot \log_c d} = (d)^{b \cdot \log_c a}\).
• Recognizing these transformations is vital for simplifying many mathematical expressions.

Understanding logarithmic properties can greatly enhance your problem-solving strategies in mathematics.

Simplification

Simplification is a key process in mathematics that makes expressions easier to handle and solve. By using known formulas and properties, such as those of logarithms, one can reduce a complicated expression to its simplest form. The goal is to reach a solution that is both clearly understandable and easy to work with. Here's how it works:

• We simplify the expression \(10^{2 \log_{10} 5}\) by applying logarithmic properties to convert it into \(5^{\log_{10} 10^2}\).
• Next, by evaluating the \(\log_{10} 10^2\) as \(2\), we transition from an exponential form to a base numerically solvable form: \(5^2\).
• The final value is a straightforward calculation: \(5 \times 5 = 25\).

This approach underscores the importance of simplifying mathematical expressions to arrive at practical and usable results.

Mathematics Education

Mathematics education is not just about learning to solve equations; it's about understanding the underlying principles and concepts. Grasping these concepts can make solving problems much less intimidating and more manageable. For example, simplifying an expression like \(10^{2 \log_{10} 5}\) prepares students for:

• Applying strategic thinking rather than rote memorization.
• Recognizing the power of properties like those of logarithms to make complex tasks simpler.
• Fostering curiosity and patience, as mathematics often involves unraveling layers of abstraction.

Ultimately, mathematics education aims to build confidence and competence in solving a wide range of problems, whether they be academic, real-world, or purely intellectual challenges.