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Chapter 12: Problem 79

Use the special properties of logarithms to evaluate each expression. \(\log _{2} 64\)

Short Answer

Expert verified
The value of \(\text{log}_{2}64\) is 6.

Step by step solution

01

Recall the Definition of Logarithms

The logarithm \(\text{log}_{b}a\) is defined as the exponent to which the base \(b\) must be raised to produce \(a\). So, \(\text{log}_{2}64\) asks us to find the power to which \(2\) must be raised to get \(64\).
02

Express 64 as a Power of 2

Recognize that \(64\) can be written as \(2^{6}\). This is because \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\).
03

Simplify Using Logarithm Properties

Using the property of logarithms that says \(\text{log}_{b}(b^k) = k\), we know \(\text{log}_{2}(2^6) = 6\).

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Key Concepts

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

Logarithms
A logarithm answers the question: 'To what exponent must a base be raised to get a certain number?' In simpler terms, a logarithm tells us how many times we need to multiply one number (the base) to reach another number. For example, \(\text{log}_{2}(8)\) means we want to know how many times we must multiply 2 to get 8. The answer is 3 because \(2 ^ 3 = 8 \). This relationship between numbers is essential for solving exponential problems.
Base
The base is the number that gets multiplied in a logarithmic expression. In the example \( \text{log}_{2} 64 \), 2 is the base. The base is always the number below the logarithm symbol. When working with logarithms, it's crucial to understand the base because it determines the value of the logarithm. If the base is 10, we call it a common logarithm, whereas a base of e (approximately 2.718) is known as a natural logarithm. In our example, since the base is 2, we need to figure out how many times 2 must be multiplied by itself to get 64.
Exponentiation
Exponentiation is the mathematical operation that increases a number (the base) by multiplying it by itself a certain number of times. The number of times we multiply the base is indicated by the exponent. For example, \(2^6\) means we multiply 2 by itself 6 times \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 \). This concept is directly related to logarithms as logarithms reverse the process of exponentiation. In \( \text{log}_{2}(64) \), we're asking how many times we need to multiply 2 to get 64. The answer is 6.
Logarithmic Simplification
Logarithmic simplification involves using properties of logarithms to make expressions easier to handle. One useful property is \( \text{log}_{b}(b^k) = k \). This means that if the argument of the logarithm is a power of the base, we can simplify it quickly. In our example, \( \text{log}_{2}(64) \) simplifies to \( \text{log}_{2}(2^6) \). According to the property, since the base and argument match, we get 6 directly. This property makes evaluating logarithms much quicker and easier.

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Most popular questions from this chapter

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ e^{\ln (6-x)}=e^{\ln (4+2 x)} $$

Use the special properties of logarithms to evaluate each expression. \(\log _{5} 1\)

Solve each equation. \(\log _{5} x=-3\)

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ e^{-0.205 x}=9 $$

Each of the following functions is one-to-one. Graph the function as a solid line (or curve), and then graph its inverse on the same set of axes as a dashed line (or curve). Complete any tables to help graph the functions. $$ f(x)=-2 x $$
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