Problem 33 - Algebra 2 and Trigonometry

Question

In \(27-56,\) evaluate each logarithmic expression. Show all work. $$ \log _{4} \frac{1}{16} $$

Step-by-Step Solution

Answer

The value of \( \log_{4} \frac{1}{16} \) is \( -2 \).

Step 1: Identify the Logarithmic Equation

We are given the expression \( \log_{4} \frac{1}{16} \). This asks us to determine the exponent to which the base 4 must be raised to yield the fraction \( \frac{1}{16} \).

Step 2: Convert to Exponential Form

The logarithmic expression \( \log_{4} \frac{1}{16} = x \) can be rewritten in exponential form as \( 4^x = \frac{1}{16} \). This step transforms the problem into an equation involving exponents.

Step 3: Express the Fraction as a Base of 4

Recognize that \( \frac{1}{16} \) can be rewritten as \( \frac{1}{4^2} \), since \( 4^2 = 16 \). Therefore, \( \frac{1}{16} = 4^{-2} \). This allows us to express the fraction in terms of the base 4.

Step 4: Solve the Exponential Equation

Set the equations equal: \( 4^x = 4^{-2} \). Since the bases are the same, the exponents must be equal, which gives us \( x = -2 \).

Step 5: Conclude the Logarithmic Evaluation

Since we found \( x = -2 \), the value of the logarithmic expression \( \log_{4} \frac{1}{16} \) is \( -2 \).

Key Concepts

Change of Base Formula

The Change of Base Formula is a helpful tool when dealing with logarithms of any base. It allows us to transform a logarithm into terms of a base that might be easier to work with, especially when using a calculator that only computes logarithms to base 10 or base \(e\). The formula is expressed as:

• \(\log_b a = \frac{\log_k a}{\log_k b}\)

Here, \(b\) and \(a\) are the original base and argument, and \(k\) is the new base, usually 10 or \(e\). To use the formula effectively:- Choose an appropriate base (like 10 or \(e\)) if you're calculating with a typical calculator.- Plug the numbers into the formula to calculate the result. Understanding this formula makes it easier to handle logarithmic expressions with different bases. It broadens your ability to calculate logarithmic expressions without directly converting to exponentials first. The Change of Base Formula is especially important when you’re working with complex calculations or need accurate results for scientific or mathematical exploration.

Exponential Form

Changing a logarithmic expression into its exponential form is a powerful tool when evaluating logarithms. The exponential form of any logarithm equation \( \log_b a = x \) is \( b^x = a \). This transformation allows us to translate a question about logs into a more familiar question about exponentials:

• The base \(b\) raised to what power \(x\) gives us the number \(a\)?

In our problem, we use this method to solve \( \log_4 \frac{1}{16} \). First, rewrite it in exponential form: \( 4^x = \frac{1}{16} \). By solving for \(x\), we can determine how many times the base needs to be multiplied by itself to produce the argument of the logarithm. This step is crucial as it simplifies logarithmic equations into something easier to understand and solve. It helps in visualizing what the logarithm really means, as exponentials are directly linked to how numbers are naturally multiplied.

Evaluating Logarithms

Evaluating logarithms is the process of determining what exponent the base must be raised to, to achieve a specific number. It's a core skill in math, enabling deeper understanding of growth patterns, exponential decay, and various scientific computations. The key steps for evaluating logarithms include:

• Identifying the base and the argument.
• Deciding whether to use the exponential form to simplify the expression.
• Applying the change of base formula as necessary, especially when the base is not immediately calculable.

When solving \( \log_4 \frac{1}{16} \), the steps involves recognizing that we need to find the exponent that makes 4 become \( \frac{1}{16} \). Converting \( \frac{1}{16} \) into an exponent with base 4, we find that it equals \(4^{-2}\). This tells us immediately that the logarithmic value is \(-2\), since we want \(4^x = 4^{-2}\) and must have \(x = -2\). This process not only answers specific homework problems but also fortifies understanding of how exponential functions relate to real-world phenomena, like interest calculations or population models.