Problem 43 - Algebra A Combined Function

Question

Find the value of each logarithmic expression. $$ \log _{4} \frac{1}{64} $$

Step-by-Step Solution

Answer

The value of \( \log_{4} \frac{1}{64} \) is \(-3\).

Step 1: Understanding the Expression

The given expression is \( \log_{4} \frac{1}{64} \). This indicates that we need to determine what power we must raise 4 to in order to get \( \frac{1}{64} \). In other words, we need to find \( x \) such that \( 4^x = \frac{1}{64} \).

Step 2: Expressing in Terms of Negative Exponent

Since the expression involves \( \frac{1}{64} \), we realize that \( 64 \) is a power of 4. Specifically, \( 64 = 4^3 \), meaning \( \frac{1}{64} = \frac{1}{4^3} = 4^{-3} \).

Step 3: Using the Definition of Logarithm

By the definition of the logarithm, if \( 4^x = 4^{-3} \), then \( x = -3 \). Thus, \( \log_{4} \frac{1}{64} = -3 \).

Key Concepts

Negative Exponents

Negative exponents can seem a bit tricky at first, but they are actually quite straightforward once you understand the concept. When you see a negative exponent, like in \( a^{-n} \), it is a way to express the reciprocal of the base raised to the positive exponent. For example:

• \( a^{-n} = \frac{1}{a^n} \)

This means that raising a number to a negative exponent "flips" the fraction. So, if you have \( 4^{-3} \), it means you have 1 over \( 4^3 \). This is crucial when solving logarithmic expressions, as we often encounter inverses like these when working with roots or fractions.
When dealing with negative exponents:

• Remember that the base remains the same, only the location in fraction flips.
• Understanding this helps solve many algebraic and calculus problems efficiently.

Logarithmic Expressions

Logarithmic expressions capture the essence of finding which power a base number must be raised to, in order to obtain a particular value. In mathematical terms, \( \log_b(y) = x \) means that \( b^x = y \). Consider the given logarithmic expression:

• \( \log_{4} \frac{1}{64} \)

Here, the task is to figure out what power of 4 results in \( \frac{1}{64} \). Logarithms provide a convenient way to reverse-extract exponents from exponential equations.
Key points to remember:

• The base of the logarithm is the number being raised to a power.
• The logarithm itself tells us the exponent needed to reach the given number.
• Understanding this concept underpins much of higher mathematics and real-world applications like computing, scientific data handling, and even sound measurement.

Powers of Numbers

The concept of powers or exponents is an essential building block in mathematics. Powers of numbers are essentially repeated multiplication. For instance, \( a^3 \) means \( a \times a \times a \). Familiarity with powers helps decode many mathematical expressions and equations.
For example, the number 64 can be expressed as a power of 4, specifically \( 4^3 \). This means multiplying 4 by itself three times results in 64.
Things to keep in mind:

• Recognizing familiar powers, such as those of 2, 3, and 4, can simplify problem-solving.
• Rewriting numbers as powers of a base is a useful technique in dividing or taking roots.
• This skill is fundamental when working with scientific calculations, algorithms, and computational problems.