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Chapter 12: Problem 41

Find the value of each logarithmic expression. $$ \log _{3} 81 $$

Short Answer

Expert verified
The value is 4.

Step by step solution

01

Identify the Base and Argument

The logarithmic expression is \( \log_{3} 81 \). Here, the base of the logarithm is 3, and the argument is 81. The expression asks for the exponent you would raise the base, 3, to in order to get 81.
02

Convert to Exponential Form

Remember that \( \log_{b}(x) = y \) implies \( b^y = x \). With \( \log_{3} 81 \), this becomes \( 3^y = 81 \).
03

Find the Exponent

Determine the power of 3 that results in 81. We know that: \[3^1 = 3, \ 3^2 = 9, \ 3^3 = 27, \ 3^4 = 81\] Hence, \( y = 4 \).
04

Verify the Calculation

Check that \( 3^4 = 81 \) is a true statement. Calculate as necessary: \( 3^4 = 3 \times 3 \times 3 \times 3 = 81 \). Since it鈥檚 correct, the logarithmic expression was solved accurately.

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Key Concepts

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

Exponential Form
When we talk about exponential form, we are essentially describing how numbers can be expressed as powers. In mathematics, this concept is fundamental and highly useful for simplifying and solving equations. For instance, any number can be expressed as a product of repeated multiplication, such as:
  • \( 3^4 \) represents 3 multiplied by itself four times: \( 3 \times 3 \times 3 \times 3 \).
  • The result of this operation is 81, which means that 81 can be written in exponential form as \( 3^4 \).
Understanding the exponential form helps us reverse the process, which is the essence of logarithms. Being able to transition smoothly between exponential and logarithmic forms allows for easier solving of math problems involving exponents.
Base of a Logarithm
In a logarithmic expression, the base is the number you repeatedly multiply. It is the number raised to a power. In the expression \( \log_{3} 81 \), 3 is the base. Understanding the base of a logarithm is crucial as it dictates the factor that is repeatedly used in multiplication.
  • For example, if the base is 3, you are working with multiples of 3.
  • The base in logarithms is the same number that you use as the multiplier in the corresponding exponential form.
Knowing how to identify and work with the logarithm's base is vital for solving logarithmic equations.
The base sets the stage for determining the exponent, a key component when interpreting the equation.
Finding Exponents
Finding exponents involves determining the power to which a base number must be raised to reach a certain value. In the context of logarithms, this means solving according to the form \( b^y = x \).
  • For the logarithmic expression \( \log_{3} 81 \), you need to find \( y \) such that \( 3^y = 81 \).
  • By breaking down calculations: \( 3^1 = 3, \ 3^2 = 9, \ 3^3 = 27, \ 3^4 = 81 \), you discover that \( y = 4 \).
This shows you the exponent of 4 is necessary for 3 to reach 81. Finding exponents is a process of matching multiplicative combinations to identify the exact power needed, which is a foundational skill in algebra.

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Most popular questions from this chapter

Let \(f(x)=\log _{5} x\). Then \(g(x)=5^{x}\) is the inverse of \(f(x)\). The ordered pair (2,25) is a solution of the function \(g(x)\). a. Write this solution using function notation. b. Write an ordered pair that we know to be a solution of \(f(x)\) c. Use the answer to part (b) and write the solution using function notation.

The formula \(y=y_{0} e^{k t}\) gives the population size y of a population that experiences a relative growth rate \(k(k\) is positive if growth is increasing and \(k\) is negative if growth is decreasing). In this formula, \(t\) is time in years and \(y_{0}\) is the initial population at time \(0 .\) Use this formula to solve Exercises 55 and \(56 .\) Round answers to the nearest year. (Source for data: U.S. Census Bureau and Federal Reserve Bank of Chicago) In \(2009,\) the population of Michigan was approximately 9,970,000 and decreasing according to the formula \(y=y_{0} e^{-0.003 t}\). Assume that the population continues to decrease according to the given formula and predict how many years after which the population of Michigan will be \(9,500,000 .\) (Hint: Let \(y_{0}=9,970,000 ; y=9,500,000\), and solve for \(t\).)

Solve. The population of the Cook Islands is decreasing according to the formula \(y=y_{0} e^{-0.0277 t}\). In this formula, \(t\) is time in years and \(y_{0}\) is the initial population at time 0 . If the size of the population in 2009 was \(11,870,\) use the formula to predict the population of Cook Islands in the year \(2025 .\) Round to the nearest whole number. (Source: The World Almanac)

Psychologists call the graph of the formula \(t=\frac{1}{c} \ln \left(\frac{A}{A-N}\right)\) the learning curve since the formula relates time t passed, in weeks, to a measure \(N\) of learning achieved, to a measure \(A\) of maximum learning possible, and to a measure c of an individual's learning style. Use this formula to answer. Round to the nearest whole number. An experiment of teaching chimpanzees sign language shows that a typical chimp can master a maximum of 65 signs. How many weeks should it take a chimpanzee to master 30 signs if \(c\) is 0.03 ?

Solve. $$ \log _{2} 8=x $$
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