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Chapter 12: Problem 76

Use the special properties of logarithms to evaluate each expression. \(12^{\log _{12} 3}\)

Short Answer

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Step by step solution

01

Recall the Property of Logarithms

The property of logarithms to use here is: if you have a number raised to the logarithm of another number with the same base, then the result is just the other number. Mathematically, this is written as: \[ a^{\text{log}_a b} = b \]
02

Apply the Property

Using the property mentioned, substitute the values into the expression. For the expression \(12^{\log_{12} 3}\), identify that the base \(a\) is 12 and \(b\) is 3. Therefore, by the property, we have: \[ 12^{\log_{12} 3} = 3 \]

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base is the number that is being multiplied by itself, and the exponent tells how many times to multiply the base. For example, in the expression \(2^3\), 2 is the base and 3 is the exponent, meaning 2 multiplied by itself three times: 2 * 2 * 2, which equals 8. Understanding exponentiation is crucial for grasping more complex mathematical concepts like logarithms.
Logarithmic Identity
A logarithmic identity is a key property or rule that logarithms follow. One of the most important identities is the one used in this exercise: \(a^{\text{log}_a b} = b\). This identity states that if you raise a number to the power of its own logarithm, you end up with the argument of the logarithm (b in this case). This identity stems from the fact that logarithms are the inverse operations of exponentiation. For any positive numbers a and b (where a 鈮� 1), this relationship holds true. This property can simplify the evaluation of seemingly complex expressions.
Base of Logarithm
The base of a logarithm is the number that is raised to a power to get another number. In the logarithmic expression \(\text{log}_a b\), 'a' is the base, and 'b' is the number being evaluated. The base is crucial because it defines the logarithmic scale and the inverse relationship with exponentiation. For example, \(\log_{10} 100 = 2\) because 10 raised to the power of 2 is 100. Different bases can be used depending on the context, such as base 10 (common logarithm), base e (natural logarithm), or any other positive number.
Evaluating Expressions
Evaluating expressions means finding their values using mathematical operations and properties. When working with logarithms and exponents, certain properties can simplify the process. For instance, in the expression \(12^{\log_{12} 3}\), the property \(a^{\text{log}_a b} = b\) can be directly applied, reducing the expression to '3'. Evaluating such expressions requires familiarity with logarithmic identities and exponentiation rules to recognize patterns and simplify calculations efficiently. This step-by-step approach ensures precision and a deep understanding of the concepts involved.

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Most popular questions from this chapter

Determine whether each statement is true or false. $$\log _{3} 49+\log _{3} 49^{-1}=0$$

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. $$3 \log _{a} 5-\frac{1}{2} \log _{a} 9$$

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. $$\left(\log _{a} p-\log _{a} q\right)+2 \log _{a} r$$

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

Solve each equation. \(\log _{6} \sqrt{216}=x\)
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