Problem 6 - Precalculus

Question

Use the property: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\) from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. \(10^{-3}=0.001\)

Step-by-Step Solution

Answer

The logarithmic form is \(\log_{10}(0.001) = -3\).

Step 1: Understand the Equation

The given equation is in the exponential form: \(10^{-3} = 0.001\). Our goal is to rewrite this exponential equation in terms of a logarithm.

Step 2: Apply Theorem 6.2

Theorem 6.2 states that \(b^{a} = c\) if and only if \(\log_{b}(c) = a\). Compare \(b=10\), \(a=-3\), and \(c=0.001\) from the given equation with the theorem.

Step 3: Rewrite in Logarithmic Form

Using Theorem 6.2, we can express the exponential equation \(10^{-3} = 0.001\) in logarithmic form as \(\log_{10}(0.001) = -3\).

Key Concepts

Understanding Exponential Functions

Exponential functions are a fundamental part of mathematics and involve expressions where a constant base is raised to a variable exponent. In the context of exponential functions, you might encounter equations like \(b^a = c\). This indicates that a base \(b\) is raised to the power \(a\), resulting in \(c\). Typically, in mathematics, exponential functions are used to model growth or decay processes, like population growth or radioactive decay.

• When the exponent \(a\) is positive, the function represents exponential growth.
• When \(a\) is negative, it represents exponential decay.

In this exercise, the given equation \(10^{-3} = 0.001\) shows exponential decay due to its negative exponent. Understanding this allows you to see why \(10\) raised to a negative power gives a small fraction (0.001).
If you are dealing with an exponential equation and need to convert it to a logarithmic form, Theorem 6.2 often comes in handy. Let's explore this further in the next section.

Navigating Through Logarithmic Equations

Logarithmic equations involve logarithms and are essentially the inverse of exponential functions. They help us deduce the power to which a number (base) must be raised to achieve another number. When you have an equation in exponential form \(b^a = c\), you can rewrite it in logarithmic form as \(\log_b(c) = a\).

• The "log base \(b\) of \(c\)" means "what power must \(b\) be raised to get \(c\)".
• This allows us to solve equations where the exponent is the unknown variable.

Using the provided equation \(10^{-3} = 0.001\), we applied Theorem 6.2, converting it to the logarithmic form \(\log_{10}(0.001) = -3\). This means that the power to which 10 must be raised to get 0.001 is \(-3\).
Understanding how to switch between exponential and logarithmic forms equips you with the tools to solve various mathematical mysteries related to growth, decay, and beyond.

Theorem 6.2: Bridging Exponents and Logarithms

Theorem 6.2 serves as a vital bridge between the world of exponential functions and logarithms. It states that \(b^a = c\) if and only if \(\log_b(c) = a\). This theorem is crucial as it allows for the transformation of exponential equations into their logarithmic counterparts, and vice versa.

• It simplifies the process of dealing with equations where finding the exponent directly is complex.
• Simplifying equations using logarithms often allows for easier manipulation and solution finding.

In the original exercise, using Theorem 6.2, the exponential equation \(10^{-3} = 0.001\) was rewritten in logarithmic form, \(\log_{10}(0.001) = -3\). It shows how interconnected these two mathematical concepts are.
By mastering this theorem, you'll not only handle similar problems more efficiently but also gain a deep understanding of the fundamental properties of functions in mathematics. This theorem exemplifies the intrinsic relationship between exponents and the logarithms, simplifying complex calculations that involve them.