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Chapter 3: Problem 2

Write an equivalent exponential equation. $$\log _{3} 81=4$$

Short Answer

Expert verified
The equivalent exponential equation is \( 3^{4} = 81 \).

Step by step solution

01

- Identify the logarithmic form

The given equation is in logarithmic form: \ \( \log_{3} 81 = 4 \). This means that the logarithm base is 3, the argument is 81, and the result is 4.
02

- Recall the definition of logarithms

The definition of a logarithm \( \log_{b} a = c \) states that \( b^{c} = a \). In this case, \( b = 3 \), \( a = 81 \), and \( c = 4 \).
03

- Rewrite the equation

Using the definition of logarithms, convert \( \log_{3} 81 = 4 \) into its exponential form: \ \( 3^{4} = 81 \).

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Key Concepts

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

Logarithmic Equations
Logarithmic equations are mathematical expressions that involve the logarithm function.
The logarithm function helps us to determine the power to which a certain number, called the base, must be raised to produce another number.
In a logarithmic equation like \( \log_{3} 81 = 4 \), the base is 3, the argument is 81, and the result is 4.
This is read as 'the logarithm of 81 with base 3 equals 4'.
These equations are useful for solving problems where the value of the exponent is unknown.
When presented with a logarithmic equation, the goal is often to convert it to an exponential equation for easier manipulation and solution.
Exponential Equations
Exponential equations involve variables in the exponent.
These types of equations take the form \( b^c = a \), where \( b \) is the base, \( c \) is the exponent, and \( a \) is the result.
For example, \( 3^4 = 81 \) shows that raising 3 to the power of 4 results in 81.
When solving an exponential equation, the objective can be to isolate the exponent or determine the base that achieves the given result when raised to the exponent.
These equations are common in growth and decay problems, such as population growth or radioactive decay.
Logarithm Definition
Understanding logarithms is key to mastering both logarithmic and exponential equations.
The logarithm of a number is the exponent to which the base must be raised to produce that number.
This relationship is written as \( \log_{b} a = c \), which is read as 'the logarithm of a with base b equals c'.
Essentially, it tells us that \( b^c = a \).
Logarithms help in solving for exponents in equations where direct computation might be challenging.
They are also pivotal in various fields like engineering, science, and finance due to their ability to simplify complex calculations.
Conversion Between Forms
Converting between logarithmic and exponential forms is a fundamental skill in algebra.
The logarithmic form \( \log_{b} a = c \) can be converted to the exponential form \( b^c = a \) and vice versa.
This process is done using the definition of logarithms, which states that the logarithm of a number is the power to which the base must be raised to generate that number.
In the provided exercise, \( \log_{3} 81 = 4 \) is converted to its exponential form \( 3^4 = 81 \), demonstrating the equivalence between the two forms.
This conversion is helpful in solving equations, especially when the logarithmic form is challenging to manipulate directly.

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Most popular questions from this chapter

The number of women graduating from 4 -yr colleges in the United States grew from \(1930,\) when 48,869 women earned a bachelor's degree, to 2005, when approximately 832,000 women received such a degree. Find an exponential function that fits the data, and the exponential growth rate, rounded to the nearest hundredth of a percent.

Write an equivalent logarithmic equation. $$10^{3}=1000$$

A quantity \(Q_{1}\) grows exponentially with a doubling time of 1 yr. A quantity \(Q_{2}\) grows exponentially with a doubling time of 2 yr. If the initial amounts of \(Q_{1}\) and \(Q_{2}\) are the same, how long will it take for \(Q_{1}\) to be twice the size of \(Q_{2} ?\)

Differentiate. $$f(x)=e^{x / 2} \cdot \sqrt{x-1}$$

Sunshine Gardens determines the following demand function during early summer for tomato plants: $$q=D(x)=\frac{2 x+300}{10 x+11}$$ where \(q\) is the number of plants sold per day when the price is \(x\) dollars per plant. (GRAPH CANNOT COPY) a) Find the elasticity. b) Find the elasticity when \(x=3\) c) At $$ 3$ per plant, will a small increase in price cause the total revenue to increase or decrease?
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