Problem 10 - 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. $$ 10^{-2}=0.01 $$

Step-by-Step Solution

Answer

\(\log_{10}(0.01) = -2\)

Step 1: Identify the Base, Exponent, and Result

Analyze the expression given: \(10^{-2} = 0.01\). Here, the base is \(10\), the exponent is \(-2\), and the result is \(0.01\).

Step 2: Write the Logarithmic Form

To convert an expression from exponential to logarithmic form, you write it as \(\log_{\text{base}}(\text{result}) = \text{exponent}\). For this example, it becomes \(\log_{10}(0.01) = -2\).

Key Concepts

Exponential to Logarithmic Conversion

Converting expressions from exponential form to logarithmic form can seem tricky at first, but it's just a matter of rearranging the parts. Let's break this down. In an exponential expression, such as \(10^{-2} = 0.01\), we have three main components. These are the base \(10\), the exponent \(-2\), and the result \(0.01\). The goal is to express this in a logarithmic form.

To convert, remember this structure: the expression \(b^x = y\) becomes \(\log_b(y) = x\). Here, \(b\) denotes the base, \(x\) is the exponent, and \(y\) is the result. So, the exponential expression \(10^{-2} = 0.01\) changes to \(\log_{10}(0.01) = -2\) in logarithmic form. Noticing this pattern makes conversion much easier. You simply switch from focusing on the power to highlighting the base's effect on the result.

With practice, converting becomes second nature. Just match the parts to their positions in logarithmic form. It's like solving a puzzle by aligning the pieces correctly.

Base and Exponent

In both exponential and logarithmic expressions, the base and exponent play central roles. Understanding their relationship is crucial for mastering these concepts. In the expression \(10^{-2} = 0.01\), \(10\) serves as the base, and \(-2\) as the exponent.

• The base is the number being multiplied.
• The exponent shows how many times the base is used in multiplication.

In exponential form, the equation illustrates multiplication or scale - the base raised to the power of the exponent equals the result.

When you convert to logarithmic form, \(\log_{10}(0.01) = -2\), it reveals the inverse relationship. Now, the expression is about finding the power \(-2\) needed for base \(10\) to produce \(0.01\).

• The base remains the same in both forms (in this case, \(10\)).
• The exponent becomes the result (\(-2\) becomes the outcome of the logarithm).

This consistent usage of base and exponent simplifies understanding growth and scale relationships when switching between forms.

Logarithmic Expressions

Logarithmic expressions might appear challenging, but their format is logical and systematic. Let's delve into \(\log_{10}(0.01) = -2\) to see how they work.

In a logarithmic expression:

• \(\log_{b}(y) = x\), where \(b\) is the base,
• \(y\) is the given number (result from the exponential form), and
• \(x\) is the logarithmic result or exponent needed to reach \(y\) from \(b\).

This accounts for why we state that the logarithm calculates the necessary exponent. In practical terms, this operation answers, "To what power must the base be raised, resulting in the given number?"

Once you understand this question, writing and reading logarithmic expressions becomes straightforward. The focus shifts from multiplying bases to understanding the exponent's impact using a logarithmic lens.

These expressions are invaluable in comparisons of growth rates, sound intensity, and more. They simplify complex multiplicative relationships into a question of scale and power.