Problem 7 - 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. $$ 2^{-2}=\left(\frac{1}{4}\right) $$

Step-by-Step Solution

Answer

\( \log_{2}(\frac{1}{4}) = -2 \).

Step 1: Identify the Base, Exponent, and Result

In the given expression, the base is 2, the exponent is -2, and the result is \(\frac{1}{4}\). This identifies the components necessary to convert the expression into logarithmic form.

Step 2: Structure the Logarithmic Form

The general form of a logarithmic equation is \( \log_{b}(x)=y \), where \(b\) is the base, \(x\) is the result, and \(y\) is the exponent. In this exercise, the base is 2, the result is \( \frac{1}{4} \), and the exponent is -2.

Step 3: Write the Logarithmic Equation

Substitute the identified values into the logarithmic form: \( \log_{2}(\frac{1}{4}) = -2 \). This converts the exponential expression into its equivalent logarithmic form.

Key Concepts

exponential expressions

Exponential expressions are mathematical phrases that involve numbers called bases raised to powers, or exponents. In mathematical terms, an expression like \(b^n\) demonstrates how many times the base \(b\) is multiplied by itself. This is a fundamental way of expressing large numbers concisely.

• The **base** is the number that's being multiplied.
• The **exponent** (or power) indicates how many times the base is multiplied by itself.
• A negative exponent means that instead of multiplying by the base, you divide by the base. For example, \(b^{-n} = \frac{1}{b^n}\).

When working with exponential expressions, understanding these components helps perform operations, such as multiplication and division of powers, with ease.

logarithms

Logarithms are the inverse operations of exponential expressions. While exponents deal with repeated multiplication, logarithms deal with finding the power or exponent itself. Converting exponential expressions into logarithmic form helps simplify calculations and solve exponential equations.
Utilizing a logarithmic form, \(\log_b(x) = y\), we interpret it as the exponent \(y\) that the base \(b\) must be raised to, in order to get the number \(x\). This is particularly useful in scientific and financial calculations where it's common to encounter data that spans wide ranges.
Here’s the structure:

• \(b\) is the **base**, which is the number raised to the power.
• \(x\) is the **result**, the number obtained when the base is raised to the exponent.
• \(y\) is the **exponent**, or the power the base is raised to reach \(x\).

Logarithms dramatically simplify complex calculations by transforming multiplication operations into addition, making them a powerful tool in mathematics.

mathematical conversions

Mathematical conversions in the context of exponential and logarithmic expressions involve changing one form into another, which is essential for solving equations more efficiently and understanding the relationships between different mathematical concepts.
To convert an exponential expression to a logarithmic form, you:

• Identify the base, which remains equivalent in both forms.
• Recognize the result of the exponentiation, which becomes the argument for the logarithm.
• Determine the exponent itself, which turns into the value that the logarithm equals.

For example, converting the exponential expression \(2^{-2} = \frac{1}{4}\) to logarithmic form involves:

• Base is 2 (the same in both expressions).
• The result is \(\frac{1}{4}\) (this becomes \(x\) in the logarithm).
• The exponent is -2 (this is \(y\) in \(\log_2(\frac{1}{4}) = -2\)).

Mastering these conversions allows you to seamlessly switch between forms, solving complex problems by choosing the most convenient mathematical representation.