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Chapter 10: Problem 176

In the following exercises, find the exact value of each logarithm without using a calculator. \(\log _{4} \frac{1}{16}\)

Short Answer

Expert verified
The exact value of \(\text{log}_{4} \frac{1}{16}\) is -2.

Step by step solution

01

Understand the Problem

To find the exact value of \(\text{log}_{4} \frac{1}{16}\), we need to determine what power the base 4 must be raised to in order to get \(\frac{1}{16}\).
02

Express as an Exponent

We start by expressing \(\frac{1}{16}\) as an exponent of 4. Notice that 16 is \(4^2\), so \(\frac{1}{16} = \frac{1}{4^2} = 4^{-2}\).
03

Use Logarithm Properties

We now use the property of logarithms which states that \(\text{log}_{b}(a^c) = c \text{log}_{b}(a)\). Therefore, the expression becomes \(\text{log}_{4}(4^{-2}) = -2 \text{log}_{4}(4)\).
04

Simplify the Expression

Since \(\text{log}_{4}(4) = 1\), it simplifies to \(-2 \times 1 = -2\). Thus, \(\text{log}_{4} \frac{1}{16} = -2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

logarithm properties
Logarithm properties play a crucial role in simplifying expressions and solving equations. One key property is the Power Rule, which states: \( \text{log}_{b}(a^c) = c \text{log}_{b}(a) \). This property was used in the example where \( \text{log}_{4} \frac{1}{16} \) was simplified. Here are some fundamental properties of logarithms:
    \( \text{log}_{b}(xy) = \text{log}_{b}(x) + \text{log}_{b}(y) \) \( \text{log}_{b}(\frac{x}{y}) = \text{log}_{b}(x) - \text{log}_{b}(y) \) \( \text{log}_{b}(x^c) = c \text{log}_{b}(x) \)
    The Power Rule was used to convert \( \text{log}_{4}(4^{-2}) \) into \( -2 \text{log}_{4}(4) \), utilizing the fact that the base of a logarithm raised to the power of the same base equals one. Other important logarithm properties include the Change of Base Formula and the fact that \( \text{log}_{b}(b) = 1 \). Understanding these properties can greatly simplify many mathematical problems.
exponents
Exponents are a way to express repeated multiplication of a number by itself. When dealing with logarithms, understanding exponents is essential. For instance, knowing that \( 4^2 = 16 \) and \( 4^{-2} = \frac{1}{16} \) helps in converting between forms.
Some key concepts about exponents to remember include:
  • Any number to the power of zero is 1: \( a^0 = 1 \)
  • Negative exponents indicate a reciprocal: \( a^{-n} = \frac{1}{a^n} \)
  • Multiplying exponents with the same base adds the powers: \( a^m \times a^n = a^{m+n} \)
  • Dividing exponents with the same base subtracts the powers: \( \frac{a^m}{a^n} = a^{m-n} \)

In the given example, recognizing that \( 16 = 4^2 \) allowed us to express \( \frac{1}{16} \) as \( 4^{-2} \), which then simplified the logarithm expression using properties of exponents.
exact value calculation
Finding the exact value of a logarithm involves understanding both logarithm properties and exponents. In the example, we found \( \text{log}_{4} \frac{1}{16} \), we:
  • Identified what power of 4 results in \( \frac{1}{16} \)
  • Expressed \( \frac{1}{16} \) as \( 4^{-2} \) using knowledge of exponents
  • Applied the Power Rule of logarithms to simplify the expression: \( \text{log}_{4}(4^{-2}) = -2 \text{log}_{4}(4) \)
  • Used the fact that \( \text{log}_{4}(4) = 1 \) to further simplify to \( -2 \times 1 = -2 \)

The result shows that understanding the steps and the core principles allows us to calculate the exact value without a calculator effectively. Practicing these concepts will help in solving more complex logarithmic problems.

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Most popular questions from this chapter

In the following exercises, solve each equation. \(\log _{4}(7 x+15)=3\)

In the following exercises, solve for \(x\). \(\log _{4} x+\log _{4} x=3\)

In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible. \(\log _{5} \sqrt[3]{x}\)

In the following exercises, solve for \(x\), giving an exact answer as well as an approximation to three decimal places. \(\left(\frac{1}{2}\right)^{x}=10\)

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places. \(3 e^{x+2}=9\)
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