Problem 23 - Introductory and Intermediate Algebra for College Students 4th

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log _{4}\left(\frac{\sqrt{x}}{64}\right)$$

Step-by-Step Solution

Answer

The expanded expression for the given logarithm is \( \frac{1}{2} \log_{4}x - 3 \)

Step 1: Convert Division to Subtraction

The quotient rule for logarithms states that the logarithm of a quotient can be written as the subtraction of the logarithms of the numerator and the denominator. So, \( \log _{4}\left(\frac{\sqrt{x}}{64}\right)\) can be rewritten as \( \log_{4}\sqrt{x} - \log_{4}64 \).

Step 2: Implement Power Rule

The power rule in logarithms states that the logarithm of a number raised to a power is equal to the product of that power and the logarithm of the number itself. So, \( \log_{4}\sqrt{x}\) can be rewritten as \( \frac{1}{2} \log_{4}x \).

Step 3: Evaluate Logarithm

The logarithm \( \log_{4}64 \) can be calculated without calculator because 64 is a power of 4. Since \(4^3 = 64\), \( \log_{4}64 = 3 \). This gives a new expression: \( \frac{1}{2} \log_{4}x - 3 \).

Step 4: Further Simplification

The final expanded logarithmic expression is \( \frac{1}{2} \log_{4}x - 3 \). Depending on the value of x, it might be possible to further simplify the expression. However, without a specific value for x, this is as much as possible to simplify the expression.