Problem 25 - Algebra A Combined Function

Question

Write each as a logarithmic equation. $$ 4^{-2}=\frac{1}{16} $$

Step-by-Step Solution

Answer

\( \log_{4}\left(\frac{1}{16}\right) = -2 \)

Step 1: Understanding the Exponential Equation

We begin with the given exponential equation: \( 4^{-2} = \frac{1}{16} \). This equation shows that the base is 4, the exponent is -2, and the result is \( \frac{1}{16} \). Our task is to express this relationship in logarithmic form.

Step 2: Identifying the Components

Recall that a logarithmic equation is the inverse of an exponential equation. The general form is \( \log_{b}(A) = x \), where \( b \) is the base of the logarithm, \( A \) is the result of the exponentiation, and \( x \) is the exponent. In our equation, \( b = 4 \), \( A = \frac{1}{16} \), and \( x = -2 \).

Step 3: Write the Logarithmic Equation

Using the components identified in the previous step, we can write the corresponding logarithmic equation as follows: \( \log_{4}\left(\frac{1}{16}\right) = -2 \). This indicates that the base 4 raised to the power of -2 gives \( \frac{1}{16} \). This is the final logarithmic form.

Key Concepts

Exponential Equations

Exponential equations involve expressions where a constant base is raised to a variable exponent. In the equation given, \( 4^{-2} = \frac{1}{16} \), we see a classic example of an exponential equation where the base is 4 and the exponent is -2. Here, the negative exponent indicates the reciprocal, which means \( 4^2 = 16 \), so \( 4^{-2} = \frac{1}{16} \). Understanding how to manipulate and transform exponential equations is key to solving them and is essential for expressing them using logarithms. Break down the equation like this:

• Identify the base: It's the number 4.
• Identify the exponent: Here, it is -2.
• Result of this exponentiation: \( \frac{1}{16} \).

By knowing these components, you can proceed to understand how to convert it to a logarithmic form.

Inverse Functions

Inverse functions essentially 'undo' each other. In the context of exponential and logarithmic equations, logarithms serve as the inverse functions of exponentials. If you have an exponential equation, you can use a logarithm to find the exponent when the other components are known. When dealing with functions, if the function \( f(x) = y \), then the inverse, \( f^{-1}(y) = x \), such as

• In exponents: \( 4^{-2} = \frac{1}{16} \) means starting with 4 raised to -2 gives \( \frac{1}{16} \).
• In logarithms: \( \log_{4}\left(\frac{1}{16}\right) = -2 \) is saying that when you take the base 4, what exponent results in \( \frac{1}{16} \)?

Thus, understanding inverse functions helps you switch between exponential and logarithmic forms.

Logarithms

Logarithms can be thought of as questions about exponents. When you see a logarithm, it essentially asks: "What power should the base be raised to, in order to obtain a certain number?" The standard notation \( \log_{b}(A) = x \) means that \( b^x = A \). This inverse relation is evident from the exponential example. Using the original problem:

• The base in the exponential equation is 4.
• The power is -2, evidenced in the logarithmic form by \( \log_{4}(\frac{1}{16}) = -2 \).
• The number \( A \) is the result of the exponential equation, \( \frac{1}{16} \).

By setting up the logarithmic equation, you are simply asking, 'To what power do we raise 4 to get \( \frac{1}{16} \)?' This power is -2.

Base of a Logarithm

The base of a logarithm is crucial because it dictates how the logarithmic relationships are structured. In the relationship \( \log_{b}(A) = x \), \( b \) is the base, \( A \) is the number you want to reach, and \( x \) is the exponent you are solving for. In our example, where the equation derived was \( \log_{4}(\frac{1}{16}) = -2 \),

• The base is 4, which is consistent throughout both exponential and logarithmic forms.
• The result \( \frac{1}{16} \) remains the output of both form's computations.
• The exponent is reflected as -2, which we extract from the logarithm.

Each part of the logarithm has a direct counterpart in the exponential equation. Understanding the base helps in connecting these forms and moving between them smoothly.