Problem 44 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$\log _{1 / 4} 64$$

Step-by-Step Solution

Answer

The simplified expression is \(-3\).

Step 1: Understand the Expression

We need to simplify the expression \( \log_{1/4} 64 \). This means we are looking for what power we need to raise \( \frac{1}{4} \) to in order to get 64.

Step 2: Use the Change of Base Formula

Recognizing that having a fractional base can make the calculation complex, let's express the logarithm using the change of base formula. We convert \( \log_{1/4} 64 \) as \( \frac{\log_{a} 64}{\log_{a} (1/4)} \) for any positive base \( a \), often we use base 10 or e (common or natural logarithms).

Step 3: Convert the Base

We choose base 4 to simplify this manually. We rewrite \( 1/4 \) as \( 4^{-1} \), and our expression becomes \( \log_{4^{-1}} 64 \). Using properties of logarithms, \( \log_{b^{-1}} x = -\log_{b} x \), so \( \log_{4^{-1}} 64 = -\log_{4} 64 \).

Step 4: Simplify the Logarithm

We simplify \( \log_{4} 64 \). Since 64 is \( 4^3 \), \( \log_{4} 64 = 3 \).

Step 5: Final Simplification

Combine the results from the previous steps. Since \( \log_{4^{-1}} 64 = -\log_{4} 64 \), and \( \log_{4} 64 = 3 \), our expression simplifies to \(-3\).

Key Concepts

Change of Base Formula

When dealing with logarithms, sometimes you'll encounter bases that aren't easy to work with. The Change of Base Formula is a nifty tool that allows you to rewrite a logarithm in terms of logs of any other base, making calculations more manageable.
\( \log_b a \) can be rewritten using any new base \( c \) as:
\[\log_b a = \frac{\log_c a}{\log_c b}\]This is particularly useful when the original base is small or a fraction, like in our original problem with base \( \frac{1}{4} \).
Using a base like 10 or \( e \), which your calculator easily handles, can simplify the computation process.

This technique highlights how logs of any base can be compared or computed using basic logarithms, like common or natural logs, effectively making the process uniform.

Properties of Logarithms

Logarithms boast several nifty properties that can simplify complex expressions. Knowing these properties is like having keys to unlock difficult problems.
Here's an essential one involving fractional bases:

• \( \log_{b^{-1}} x = -\log_b x \) - This reveals how flipping the base impacts the sign of the log.

In our exercise, using the property \( 1/4 \) rewritten as \( 4^{-1} \), gives us great leverage.
Also, remember these other essential properties:

• \( \log xy = \log x + \log y \) - useful for multiplication inside a log.
• \( \log \frac{x}{y} = \log x - \log y \) - handy for division within a log.
• \( \log x^n = n \cdot \log x \) - great for powers inside a log. Perfect for our simplification here.

Mastering these can drastically simplify tasks like determining powers or complex divisions inside logarithms.

Exponents

In mathematics, logarithms and exponents are two sides of the same coin. Understanding exponents is fundamental when working with logarithms.
An exponent tells you how many times to multiply a number, called the base, by itself. For instance, \( 4^3 \) equals 64, meaning we multiply 4 by itself 3 times.

This concept is critical in solving logarithmic expressions. In reverse, when a logarithm asks what power a base must be raised to reach a certain number (like in \( \log_4 64 = 3 \)), we derive the meaning of the exponent: it's 3.

• If you see \( x^a \), you interpret it as: Multiply \( x \) by itself \( a \) number of times.
• Understanding exponents is crucial for creating equivalences in logarithm problems.

Remember, grasping the flip side of exponents (logarithms) can give more insight into their power and application.