Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-14,\) write each exponential equation in logarithmic form. $$ 0.001=100^{-\frac{3}{2}} $$

Step-by-Step Solution

Answer

\(-\frac{3}{2} = \log_{100} 0.001\).

Step 1: Identify the Exponential Equation Components

An exponential equation is of the form \(a = b^c\). In our case, \(a = 0.001\), \(b = 100\), and \(c = -\frac{3}{2}\).

Step 2: Understand the Logarithmic Form

The logarithmic form answers the question: 'To what power must we raise the base to get the given number?' It is written as \( c = \log_b a \).

Step 3: Rewriting in Logarithmic Form

Using the values identified: \(a = 0.001\), \(b = 100\), and \(c = -\frac{3}{2}\), the logarithmic form is: \(-\frac{3}{2} = \log_{100} 0.001\).

Key Concepts

Exponential Equations

An exponential equation is an equation in which a constant base is raised to a variable exponent. Consider this form:

• General form: \(a = b^c\)
• Components: \(a\) (result), \(b\) (base), \(c\) (exponent)

These equations are commonly seen in growth and decay processes.

Exponential equations enable us to model scenarios where quantities grow or shrink rapidly.
To solve these, observe the relationships among the base, exponent, and the result.
Understand the roles of these components to change between exponential and logarithmic forms.

Converting Exponentials to Logarithms

Changing an exponential statement into logarithmic form helps us understand the relationships within an equation.

Follow these steps to convert:

• Identify the equation structure (\(a = b^c\))
• Flip it to determine the unknown exponent: (\(c = \log_b a\))

By rewriting the statement, you're asking: "To what power must the base \(b\) be raised to yield \(a\)?"
This conversion is beneficial for solving problems that involve unknown exponents and for simplifying complex equations.

Logarithmic Form

Once an exponential equation is transformed, it appears in logarithmic form. This new format reveals unique insights:

• Formula: \(c = \log_b a\)
• Reading: "\(c\) is the power to which base \(b\) must be raised to obtain \(a\)."

The goal of using logarithmic form is to simplify solving for unknown variables.

In our exercise, rewriting \(0.001 = 100^{-\frac{3}{2}}\) yields \(-\frac{3}{2} = \log_{100} 0.001\).
This confirms that \(-\frac{3}{2}\) is the exponent needed to make base \(100\) equal to \(0.001\). By understanding and utilizing this form, further complexities in mathematics can be resolved with ease.