Problem 30 - A Graphical Approach to College Algebra

Question

Solve each equation. Give the exact answer. $$\log _{1 / 4} \frac{16^{2}}{2^{-3}}=x$$

Step-by-Step Solution

Answer

The exact answer is \( x = -\frac{11}{2} \).

Step 1: Understand the Base Change

The exercise gives us a logarithmic equation involving base \( \frac{1}{4} \). Recognize that \( \frac{1}{4} = 4^{-1} \). We will use this to help simplify and solve the equation.

Step 2: Simplify the Argument of the Logarithm

The argument inside the logarithm is \( \frac{16^2}{2^{-3}} \). We can simplify this by evaluating \( 16^2 \) and \( 2^{-3} \):\[ 16^2 = (2^4)^2 = 2^8 \]\[ 2^{-3} = \frac{1}{2^3} \]Therefore, \( \frac{16^2}{2^{-3}} \) becomes:\[ \frac{2^8}{2^{-3}} = 2^{8-(-3)} = 2^{11} \]

Step 3: Rewrite the Logarithm Equation

We insert \( 2^{11} \) back into the original logarithm equation:\[ \log_{1/4}(2^{11}) = x \]Recall that \( \log_{a}(b^c) = c \cdot \log_{a}(b) \). This allows us to rewrite the equation as:\[ 11 \cdot \log_{1/4}(2) = x \]

Step 4: Change of Base Formula

To find \( \log_{1/4}(2) \), use the change of base formula: \[ \log_{1/4}(2) = \frac{\log_{2}(2)}{\log_{2}(1/4)} \]We know \( \log_{2}(2) = 1 \) and \( 1/4 = 2^{-2} \), so:\[ \log_{2}(1/4) = \log_{2}(2^{-2}) = -2 \times \log_{2}(2) = -2 \]This gives us:\[ \log_{1/4}(2) = \frac{1}{-2} = -\frac{1}{2} \]

Step 5: Solve for x

Now substitute back into the expression we got from Step 3:\[ 11 \cdot \left(-\frac{1}{2}\right) = x \]Calculate the product:\[ x = -\frac{11}{2} \]

Key Concepts

Change of Base Formula

When working with logarithmic equations, you might encounter a base that isn't convenient for calculation. This is where the **Change of Base Formula** comes into play. It allows you to switch the base of a logarithm to another base, typically to the base 10 or base 2. The formula is:\[\log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)}\]This formula is especially useful when dealing with complex logarithms on calculators that do not support arbitrary bases. For example, to solve \( \log_{1/4}(2) \), you can use the change of base formula by choosing a base that's easier to compute, like base 2. This helps simplify the calculation and provides an exact solution. Always remember to evaluate the new, simpler logarithms with the base of your choice, as this provides greater ease and flexibility in solving logarithmic equations. This strategy is instrumental in converting a challenging problem into a more manageable computation.

Exponentiation

Exponentiation is a fundamental mathematical operation where a number, known as the base, is multiplied by itself a specified number of times. The number of times it is multiplied is called the exponent. For instance, if you have \( 2^3 \), it means you multiply 2 by itself three times: \( 2 \times 2 \times 2 = 8 \).In the context of logarithmic equations, understanding exponentiation is crucial because logarithms are inherently related to powers of numbers. In our example, \( 16^2 \) can be represented as \((2^4)^2 = 2^8\). Recognizing how to convert and simplify such expressions using exponentiation and their properties, like the power of a power rule, is key to handling fractional and negative exponents such as \(2^{-3} = \frac{1}{2^3}\).Essentially, exponentiation helps reduce complex expressions to simpler ones and aids in solving logarithmic equations, where you consistently turn complicated equations into basic power terms for further simplification.

Simplification of Expressions

Simplifying expressions involves reducing them to their simplest form to ease calculations and understand the underlying values better. It's a necessary step in mathematics, especially when solving equations.To simplify expressions like \( \frac{16^2}{2^{-3}} \), apply laws of exponents such as:

• \( a^m/a^n = a^{m-n} \)
• \( (a^m)^n = a^{m \times n} \)
• \( a^{-n} = 1/a^n \)

Using these rules, simplify \( \frac{2^8}{2^{-3}} \) to \( 2^{8 - (-3)} = 2^{11} \). This simplification allows us to evaluate logarithmic expressions more effectively by breaking down complex parts into their foundational components.The goal of simplification is to convert a difficult problem into an understandable form so that further operations, like applying logarithmic properties, become more straightforward. Always look to simplify by reducing fractions, factoring when possible, and ultimately aiming to express components in terms of their simplest power forms.