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Chapter 12: Problem 67

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

Short Answer

Expert verified
\( \log_{3} 3 = 1 \)

Step by step solution

01

Identify the Base and Argument

The given expression is \( \log_{3} 3 \). Here, the base of the logarithm is 3, and the argument is also 3.
02

Use the Logarithm Property

Recall the property of logarithms that states \( \log_{b} b = 1 \). This property holds because any number raised to the power of 1 equals itself.
03

Apply the Property

Using the property mentioned, \( \log_{3} 3 = 1 \).

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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 the base be raised, to produce a given number? For example, in the logarithm \(\text{log}_{b}a\), 'b' is the base and 'a' is the argument. The result of the logarithm is the exponent 'x' such that \({b^x = a}\).
When we encounter \(\text{log}_{3} 3\), it is asking: '3 raised to what power equals 3?'
  • In this scenario, the exponent is 1 because \({3^1 = 3}\).
  • Therefore, \(\text{log}_{3} 3 = 1\).
Logarithms are incredibly useful in solving equations with exponents, understanding growth patterns, and performing calculations in various fields like science and engineering.
Base and Argument
In a logarithmic expression, the two main components we need to focus on are the base and the argument.
Here鈥檚 a simple way to remember them:
  • The 'base' is the number that is being raised to a power. It is written as a subscript.
  • The 'argument' is the number we are trying to represent as a power of the base. It is written next to the base but not as a subscript.
For example, in \(\text{log}_{3} 3\):
  • The 'base' is 3.
  • The 'argument' is also 3.
This tells us that we need to find the exponent to which 3 must be raised to result in 3. In this case, the answer is 1, because \({3^1 = 3}\).
Understanding the roles of the base and argument helps in simplifying and solving logarithmic expressions accurately.
Logarithm Property
There are several properties of logarithms that make it easier to work with them. One such common property is:
\(\text{log}_{b} b = 1\).
  • This property states that any number raised to the power of 1 is the number itself.
  • In general, \(\text{log}_{b} b = 1\), for any base 'b'.

Applying this property to our example: In \(\text{log}_{3} 3\), the base is 3 and the argument is also 3. By using the property \(\text{log}_{b} b = 1\):
  • We know that \(\text{log}_{3} 3 = 1\)
. Recognizing and utilizing this property simplifies solving logarithmic equations. It is especially useful in quick evaluations and calculations, saving both time and effort.

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Most popular questions from this chapter

Solve each equation. \(\log _{x} 5=\frac{1}{2}\)

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ \ln e^{3 x}=9 $$

Determine whether each statement is true or false. $$\log _{2}(8+32)=\log _{2} 8+\log _{2} 32$$

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. $$\frac{1}{3} \log _{b} x+\frac{2}{3} \log _{b} y-\frac{3}{4} \log _{b} s-\frac{2}{3} \log _{b} t$$

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. $$\log _{10}(x+3)+\log _{10}(x+5)$$
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