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Chapter 10: Problem 82

Use the special properties of logarithms to evaluate each expression. $$\log _{3} 27$$

Short Answer

Expert verified
\(\log_{3} 27 = 3\)

Step by step solution

01

Understand the Problem

The problem requires evaluating the expression \(\log_{3} 27\) using the properties of logarithms.
02

Use the Definition of Logarithms

Recall that \(\log_b(a) = c\) means \(b^c = a\). In this case, we want to find a value of \(c\) such that \(3^c = 27\).
03

Express 27 as a Power of 3

Notice that \(27\) can be written as a power of \(3\). Specifically, \(27 = 3^3\).
04

Evaluate the Logarithm

Since \(27 = 3^3\), we can replace \(27\) with \(3^3\) in the logarithm expression: \(\log_{3}(3^3)\).
05

Apply the Power Rule of Logarithms

Using the power rule of logarithms, \(\log_b(b^c) = c\), we have \(\log_{3}(3^3) = 3\).

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Key Concepts

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

logarithms
A logarithm is a mathematical function that helps us solve for exponents. Imagine you have an equation like this: $$3^x = 9$$, and you want to find the value of $$x$$. A logarithm can help you with that! The logarithmic form of this equation would be $$\text{log}_3(9) = x$$. Here, $$3$$ is the base of the logarithm, $$9$$ is the number you are finding the log of, and $$x$$ is the exponent to which the base must be raised to get the number. This means that if you raise $$3$$ to the power of $$2$$, you get $$9$$, so $$\text{log}_3(9)$$ equals $$2$$.
power rule of logarithms
The power rule of logarithms is a handy property that simplifies expressions involving logarithms of powers. It states that $$\text{log}_b(a^c) = c \times \text{log}_b(a)$$. This rule can make complex logarithmic expressions much simpler. For example, let's say you have $$\text{log}_2(8)$$. You might recognize that $$8$$ is the same as $$2^3$$. Using the power rule, you can rewrite the expression as $$3 \times \text{log}_2(2)$$. Since $$\text{log}_2(2) = 1$$, the expression simplifies to $$3 \times 1 = 3$$. This property is particularly useful when dealing with logarithms of numbers that can be expressed as powers of the base.
definition of logarithms
The definition of logarithms is crucial for understanding more complex properties. Logarithms are the inverse operations of exponentiation. In other words, if you have an equation in the form $$b^c = a$$, you can use logarithms to find $$c$$ by rewriting the equation as $$\text{log}_b(a) = c$$. For instance, if you have $$2^3 = 8$$, you can find that $$\text{log}_2(8) = 3$$. This definition helps convert between exponential and logarithmic forms of equations, making it easier to solve equations involving exponents. Using the definition, you can tackle problems like finding the unknown exponent, base, or the number in many mathematical contexts.

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Most popular questions from this chapter

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

Solve each equation. Give exact solutions. $$ \log _{2}\left(x^{2}+7\right)=4 $$

Based on selected figures obtained during the years \(1970-2015,\) the total number of bachelor's degrees earned in the United States can be modeled by the function $$ D(x)=792,377 e^{0.01798 x} $$ where \(x=0\) corresponds to \(1970, x=5\) corresponds to \(1975,\) and so on. Approximate, to the nearest unit, the number of bachelor's degrees earned in 2015. (Data from U.S. National Center for Education Statistics.)

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. $$ \log _{3} \sqrt{2} $$

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. $$ \log _{\pi} e $$
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