Problem 5 - Intermediate Algebra

Question

Write each as an exponential equation. See Example 1. $$ \log _{10} 1000=3 $$

Step-by-Step Solution

Answer

The exponential equation is \( 10^3 = 1000 \).

Step 1: Identify the Components of the Logarithm

The given logarithmic equation is \( \log_{10} 1000 = 3 \). Here, 10 is the base of the logarithm, 1000 is the result or what the logarithm is applied to, and 3 is the logarithm's output or power.

Step 2: Rewrite Using Exponential Form

The rule for converting a logarithmic equation to an exponential form is: if \( \log_a b = c \), then \( a^c = b \). Using this rule, convert \( \log_{10} 1000 = 3 \) to its exponential form: \( 10^3 = 1000 \).

Step 3: Verify the Exponential Equation

Verify that the exponential equation \( 10^3 = 1000 \) is correct by calculating that 10 to the power of 3 indeed equals 1000. Multiply 10 by itself twice: \( 10 \times 10 = 100 \), then \( 100 \times 10 = 1000 \). This confirms the correctness of the conversion.

Key Concepts

Logarithms

Logarithms are a fundamental concept in mathematics, acting as the inverse operation to exponentiation. Imagine you have a power equation like this: you know the result of raising a number to a certain power, and you need to find what that power is. That’s where logarithms come in. They help you "work backwards" from the result to discover the exponent. In the expression \( \log_{10} 1000 = 3 \), the logarithm \( \log_{10} \) indicates that 10 is the base.
The base of the logarithm tells you what number you’re repeatedly multiplying. Here, it is 10 because \( 10 \times 10 \times 10 = 1000 \). Logarithms make complex multiplication tasks easier, especially when dealing with large numbers.

• Base (10 in this case): The number being multiplied.
• Argument (1000 here): The result of the multiplication.
• Logarithmic Value (3): The number of times you multiply the base by itself.

Understanding logarithms helps you simplify and solve problems involving exponential growth, compound interest, and more.

Base and Exponent

In exponential equations, understanding the base and exponent is crucial. The base is the number that gets multiplied over and over. The exponent tells you how many times to multiply the base by itself. In simple terms, if you have an equation like \( a^b = c \),

• \( a \) is the base,
• \( b \) is the exponent, and
• \( c \) is the result.

In the example \( 10^3 = 1000 \), 10 is the base, 3 is the exponent, and 1000 is the result.
Each part of an exponential equation plays a role in defining the relationship between the numbers:

• The base determines what number you're using to build up.
• The exponent tells you how many times to use the base in the multiplication.
• The result shows you the outcome of this repeated multiplication.

Mastering the base and exponent allows you to solve and understand more complex mathematical expressions.

Converting Logarithmic Equations

Converting logarithmic equations into exponential form is a key skill in mathematics. This conversion helps make sense of equations that involve logarithmic expressions, allowing you to see the step-by-step power relationships. The conversion process follows the rule: \( \log_a b = c \) converts to \( a^c = b \).
Let’s break it down using our example \( \log_{10} 1000 = 3 \):

• The base (10) becomes the base of the exponential equation.
• The logarithm's result (3) is the exponent.
• The argument (1000) is now the result of the exponential equation.

By transforming these forms, you transform a question of "what power can achieve this result?" into "this power achieves this result." This not only aids in computation but also deepens your comprehension of mathematical relationships. Understanding this conversion strengthens your ability to swing between logarithms and exponential expressions fluently.

Mathematics Education

Mathematics education continually emphasizes the importance of understanding core concepts like logarithms and exponents because they form the backbone of advanced mathematics and real-life applications. By learning these concepts, you develop critical thinking and problem-solving skills that go beyond simple arithmetic or rote memorization to decode patterns and relationships in numbers.
In practice:

• Logarithms help manage data growth in fields such as computer science and information theory.
• Exponents and powers are essential in scientific equations and engineering designs.
• Mastering conversion between different mathematical forms makes complex problems easier.

A solid grasp on these areas boosts your confidence in tackling real-world problems where mathematical logic is required.
From calculating interest rates to predicting population growth, a proper mathematics education equips you with tools to engage with and solve everyday challenges analytically.