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Chapter 12: Problem 23

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

Short Answer

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

Step by step solution

01

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 \( \log_{4}\sqrt{x} - \log_{4}64 \).
02

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 \( \frac{1}{2} \log_{4}x \).
03

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: \( \frac{1}{2} \log_{4}x - 3 \).
04

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.

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Most popular questions from this chapter

Complete the table for a savings account subject to continuous compounding ( \(A=P e^{n}\) ). Round answers to one decimal place. $$\begin{array}{l|c|l|c} \hline \begin{array}{l} \text { Amount } \\ \text { Invested } \end{array} & \begin{array}{l} \text { Annual Interest } \\ \text { Rate } \end{array} & \begin{array}{l} \text { Accumulated } \\ \text { Amount } \end{array} & \begin{array}{l} \text { Time } t \\ \text { in Years } \end{array} \\ \hline \$ 2350 & & \text { Triple the amount invested } & 7 \\ \hline \end{array}$$

Solve each equation in Exercises \(144-146 .\) Check each proposed solution by direct substitution or with a graphing utility. $$\ln (\ln x)=0$$

Logarithmic models are well suited to phenomena in which growth is initially rapid, but then begins to level off. Describe something that is changing over time that can be modeled using a logarithmic function.

Without using a calculator, determine which is the greater number: \(\log _{4} 60\) or \(\log _{3} 40\)

Solve: \(x+3 \leq-4\) or \(2-7 x \leq 16\) (Section 9.2, Example 6)
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