Problem 4 - College Algebra and Calculus: An Applied Approach

Question

Match the logarithmic equation with its exponential form. [The exponential forms are labeled (a), (b), (c), (d), (e), and (f).]\(\log _{4} \frac{1}{16}=-2 \quad\) (d) \(4^{-2}=\frac{1}{16}\)

Step-by-Step Solution

Answer

The logarithmic equation \(log _{4} \frac{1}{16}=-2\) matches with the exponential equation \(d) : 4^{-2} = \frac{1}{16}\)

Step 1: Understand the relationship between logarithmic and exponential equations

Remember that the logarithmic function \(log_b a = c\) is equivalent to its exponential form \(b^c = a\).

Step 2: Apply the relationship to the given problem

Given the logarithmic equation \(log _{4} \frac{1}{16} = -2\), apply the relationship between logarithms and exponentials. This gives the exponential equation \(4^{-2} = \frac{1}{16}\)

Step 3: Match the obtained exponential equation with the options given

Find the equal exponential equation among the given options. Since \(4^{-2} = \frac{1}{16}\) matches with the option \(d\) : \(4^{-2}=\frac{1}{16}\), therefore, option \(d\) is the correct match.

Key Concepts

Logarithmic Functions

A logarithmic function can be thought of as the opposite of an exponential function.
Just like subtraction undoes addition, logarithms undo exponentiation. When we say \ \( \log_{b} a = c \ \), we mean that \ \( b^c = a \ \).
The value of a logarithm tells us the power we must raise the base \( b \) to produce \( a \).
For example, in the equation \ \( \log_{4} \frac{1}{16} = -2 \ \), it signifies that 4 raised to the power of -2 results in \( \frac{1}{16} \).

Some important points to remember about logarithms:

• If there's no base written, it's assumed to be 10 (called the common log).
• A natural logarithm has a base of \( e \), approximately 2.718.
• Logarithms simplify the multiplication of large numbers into simple addition operations.

By understanding the basics of logarithms, you can tackle a range of math problems with more confidence and ease.

Exponential Functions

Exponential functions are central in mathematics because they show how quantities grow or shrink over time.
In an exponential function like \( b^x \), \( b \) is the base and \( x \) is the exponent. The base \( b \) is a constant, while \( x \) can be any real number.
This function describes a constant rate of proportional growth or decay.

Consider the exponential equation from earlier: \( 4^{-2} = \frac{1}{16} \). This example shows an exponential decay because of the negative exponent.

• With positive exponents, the function grows rapidly as \( x \) increases.
• With negative exponents, the function approaches zero, demonstrating decay.
• If the base is greater than 1, the function grows; if between 0 and 1, it decays.

Exponential functions appear in many fields, such as biology for modeling population growth, or finance for calculating compound interest.
Mastering them is crucial for solving real-world problems.

Equation Conversion

Conversion between logarithmic and exponential forms is a valuable skill.
It allows you to move easily between the two forms depending on what the problem requires.
For the given problem, recognizing the relationship lets us rewrite \( \log_{4} \left( \frac{1}{16} \right) = -2 \) as \( 4^{-2} = \frac{1}{16} \).

Here's a step-by-step on how to convert between these forms:

• Identify the base \( b \), the argument \( a \), and the logarithm result \( c \).
• Recall that \( \log_{b} a = c \) can be expressed as \( b^c = a \).
• Rearrange the terms based on the format you need. For logarithms, simplify by putting the base as the denominator after exponentiating for the exponential form.

This skill not only simplifies calculations but also enhances understanding of inverse functions, making complex problems more manageable.