Problem 9 - Intermediate Algebra

Question

For Problems \(1-10\), write each of the following in logarithmic form. For example, \(2^{3}=8\) becomes \(\log _{2} 8=3\) in logarithmic form. $$ 10^{-1}=0.1 $$

Step-by-Step Solution

Answer

\(\log_{10} 0.1 = -1\)

Step 1: Understand Exponential Form

The given expression is in exponential form, which can be identified by the format: base\(^{\text{exponent}} = \text{result}\). In this case, the base is 10, the exponent is -1, and the result is 0.1. This is written as \(10^{-1} = 0.1\).

Step 2: Recall Logarithmic Form Structure

The general structure for writing in logarithmic form is \(\log_{\text{base}} \text{result} = \text{exponent}\), where you solve for the exponent that the base is raised to in order to achieve the result.

Step 3: Apply to Given Expression

For the equation \(10^{-1} = 0.1\), translate this into logarithmic form as follows: identify 10 as the base, 0.1 as the result, and -1 as the exponent. Therefore, write \(\log_{10} 0.1 = -1\).

Key Concepts

Exponential Expression

An exponential expression involves a base and an exponent and is written in the form: base\(^{\text{exponent}} = \text{result}\). This signifies that the base is multiplied by itself as many times as indicated by the exponent. In simpler terms, the exponent shows how many times you use the base in a multiplication.
It's like a shortcut for multiplication. For example, in the expression \(10^{-1} = 0.1\), 10 is the base and -1 is the exponent. This means you take 10 and multiply it "-1 times" or, more accurately in this case, take the reciprocal of 10.
Exponents can be positive or negative which affects how you perform the calculation:

• A positive exponent, like 3 in \(2^3\), means multiply the base by itself three times (\(2 \times 2 \times 2 = 8\)).
• A negative exponent, like -1 in \(10^{-1}\), represents the reciprocal of the base raised to the corresponding positive exponent (\(1/10^1 = 0.1\)).

By grasping the way an exponential expression functions, you can get more comfortable with using it in different mathematical contexts.

Base and Exponent

In mathematics, the terms 'base' and 'exponent' are the foundation of an exponential expression. Understanding these two components will help you interpret and solve expressions in both exponential and logarithmic forms.

• The **base** of an expression is the number that is getting multiplied. It's the main workhorse of the expression. In \(10^{-1}\), 10 is the base. It shows you which number is being repeatedly used.
• The **exponent** tells you how many times the base is used in the multiplication. Exponents can also represent fractions, indicating roots, or negative values, pointing to reciprocals.

When you encounter expressions like \(b^{n} = a\), note that:

• \(b\) is the base.
• \(n\) is the exponent.
• \(a\) is the result of multiplying the base \(b\) by itself \(n\) times (or calculating its reciprocal, root, depending on the exponent).

Every exponential expression tells a story of the base. By comprehending how bases and exponents interact, solving such mathematical problems becomes significantly easier.

Mathematics Education

Teaching and learning mathematics often involve transforming problems from one form to another, testing your understanding of fundamental concepts. A pivotal conversion is between exponential and logarithmic forms.
In mathematics education, recognizing this relationship enhances problem-solving skills. When turning exponential expressions into logarithmic form, you're learning a new perspective of viewing a relationship between numbers. This is crucial in higher math where you delve into areas such as calculus and algebra.
Let's simplify: When you see \(10^{-1} = 0.1\) and convert it to \(\log_{10} 0.1 = -1\), you're switching from thinking "ten to the negative first power equals zero point one" to "what power do I raise ten to get zero point one?"
This switch fosters a deeper mathematical understanding and develops critical thinking. In education:

• Interactive learning, like visual aids and online simulations, makes such transitions clearer.
• Group activities can also help students share different solving strategies.

These educational strategies contribute greatly to understanding the beauty and functionality of mathematics, making such concepts less daunting.