Problem 13 - Algebra 2

Question

Write each equation in logarithmic form. $$ 10^{-2}=0.01 $$

Step-by-Step Solution

Answer

The logarithmic form of the equation \(10^{-2}=0.01\) is \(\log_{10}(0.01) = -2\).

Step 1: Understand Exponential Form

Consider the given exponential equation \(10^{-2} = 0.01\). One needs to understand the components here. The base of the given equation is 10, the exponent is -2, and the result is 0.01.

Step 2: Convert to Logarithmic Form

In an exponential equation \(b^y = x\), b is the base, y is the exponent and x is the result. The equation can be converted into logarithmic form as \(\log_b(x) = y\). Using this form, the provided equation \(10^{-2} = 0.01\) can be written in logarithmic form as \(\log_{10} (0.01) = -2\).

Key Concepts

Exponential Equations

Understanding exponential equations is crucial for grasping many advanced mathematical concepts. An exponential equation is a mathematical expression where a constant base is raised to a variable exponent. The general form is represented as \( b^y = x \). Here, \( b \) is the base, \( y \) is the exponent, and \( x \) is the result.

Examples of exponential equations include real-world situations like population growth, radioactive decay, or even calculating compound interest. In our exercise, \( 10^{-2} = 0.01 \) is a simple exponential equation.

This tells us that when the base 10 is raised to the power of -2, the result is 0.01. Remembering this framework helps us transition to understanding how these equations can be expressed in logarithmic form.

Logarithmic Form

The logarithmic form is another way to express an exponential equation. It helps in solving equations where the unknown is an exponent. The key conversion from an exponential form \( b^y = x \) to logarithmic form is given by \( \log_b(x) = y \). Each component of the exponential equation is translated into the logarithmic form.

• \( b \) becomes the base of the logarithm.
• \( y \), the exponent, becomes the result of the logarithmic function.
• \( x \) is placed inside the logarithmic function as the argument.

In the example \( 10^{-2} = 0.01 \), the base is 10, the exponent is -2, and the result is 0.01. Converting this into logarithmic form, we have \( \log_{10}(0.01) = -2 \). This shows that the power to which 10 must be raised to yield 0.01 is -2.

Logarithmic Conversion

Converting between exponential and logarithmic forms is a fundamental skill in algebra. Logarithmic conversion helps in solving equations where it's necessary to isolate the variable. By understanding both forms, switching between them becomes easier and more intuitive.

The process involves identifying the base, exponent, and result in the exponential form and then rearranging these into the logarithmic form. Let's take a look at a simple process of conversion:

• Identify the base "\( b \)" in the exponential form \( b^y = x \).
• Determine the result "\( x \)" and the exponent "\( y \)".
• Formulate the logarithmic equation as \( \log_b(x) = y \).

For instance, in our example \( 10^{-2} = 0.01 \), the base is 10, the result is 0.01, and the exponent is -2. Thus, rewriting it logarithmically gives us \( \log_{10}(0.01) = -2 \). This conversion aligns the understanding of both exponential growth and decay and provides a deeper insight into the values being calculated.