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Chapter 4: Problem 7

Express the equation in exponential form. (a) \(\ln 5=x\) (b) \(\ln y=5\)

Short Answer

Expert verified
(a) \( e^x = 5 \), (b) \( y = e^5 \).

Step by step solution

01

Understanding the Natural Logarithm

The natural logarithm function, denoted by \( \ln \), is the inverse of the exponential function with base \( e \). This means that if \( \ln(a) = b \), then \( a = e^b \). Our goal is to convert the given logarithmic equations into their equivalent exponential form.
02

Solving part (a) \( \ln 5 = x \)

To express the equation \( \ln 5 = x \) in exponential form, we need to use the property of the natural logarithm. Using the definition, we rewrite \( 5 \) as an exponential function: \[ e^x = 5 \]This is the exponential form of the given logarithmic equation.
03

Solving part (b) \( \ln y = 5 \)

Similarly, for the equation \( \ln y = 5 \), we again use the property of natural logarithms to write it in exponential form: \[ y = e^5 \]This is the exponential form of the second given logarithmic equation.

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Key Concepts

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

Natural Logarithm
The natural logarithm, represented by \( \ln \), is a special type of logarithm where the base is the number \( e \). This particular logarithm measures the power to which the base \( e \) must be raised to obtain a given number. In essence, when you see \( \ln(a) = b \), it indicates that the base \( e \) raised to the power \( b \) will give you \( a \). This relationship between the base \( e \), the natural logarithm, and exponential growth is fundamental in mathematics and especially in calculus.
  • The base \( e \) is approximately equal to 2.71828, and it's a cornerstone of continuous growth processes.
  • The natural logarithm is commonly used in solving equations involving exponential functions because it simplifies the process.
When working with natural logarithms, you often perform operations to transition between logarithmic and exponential forms, as shown in the solutions to the exercise.
Inverse Function
In mathematics, the concept of an inverse function is crucial. An inverse function essentially "reverses" the effect of a function. If you start with a number, apply a function \( f \), and then apply its inverse \( f^{-1} \), you should end up back where you started.
  • In terms of natural logarithms and exponential functions, the natural logarithm is the inverse of the exponential function when the base is \( e \).
  • This means when you have \( e^b = a \), applying \( \ln \) to \( a \) will give you \( b \).
This inverse relationship allows for the conversion between exponential and logarithmic equations, which is exactly what's happening in the original exercise when equations are expressed in exponential form from logarithmic form.
Exponential Function
The exponential function is a mathematical concept where a constant base is raised to the power of a variable. In the context of this exercise, we focus on the exponential function with base \( e \), denoted as \( e^x \). This function is integral in describing situations where there is continuous growth or decay, such as in populations, finance, and natural phenomena.
  • For example, the equation \( e^x = 5 \) from the exercise is an exponential equation reflecting growth.
  • Understanding how to express logarithmic equations in this form is important for solving complex mathematical problems involving growth and decay.
Overall, the exponential function is a pivotal tool used in real-world applications and theoretical mathematics.
Base e
The base \( e \) is a mathematical constant that roughly equals 2.71828 and is the foundation of natural logarithms. This number is essential in the study of continuous compounding and growth processes. The significance of base \( e \) arises because it is involved in growth rates and helps model complex, continuous processes in mathematics.
  • The number \( e \) was discovered in the context of compound interest in finance, where it helped simplify calculations when interest is added continuously.
  • The exponential function, with base \( e \), \( e^x \), is crucial for calculations involving natural logarithms because they are inverse operations.
With its unique properties, base \( e \) has found applications in various fields including engineering, biology, economics, and beyond, making it an invaluable tool for scientists and mathematicians alike.

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Most popular questions from this chapter

These exercises deal with logarithmic scales. The hydrogen ion concentrations in cheeses range from \(4.0 \times 10^{-7} \mathrm{M}\) to \(1.6 \times 10^{-5} \mathrm{M}\). Find the corresponding range of pH readings.

The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date. (a) Find the present value of \(\$ 10,000\) if interest is paid at a rate of \(9 \%\) per year, compounded semiannually, for 3 years. (b) Find the present value of \(\$ 100,000\) if interest is paid at a rate of \(8 \%\) per year, compounded monthly, for 5 years.

These exercises deal with logarithmic scales. The Northridge, California, earthquake of 1994 had a magnitude of 6.8 on the Richter scale. A year later, a 7.2-magnitude earthquake struck Kobe, Japan. How many times more intense was the Kobe earthquake than the Northridge earthquake?

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When a certain medical drug is administered to a patient, the number of milligrams remaining in the patient's bloodstream after \(t\) hours is modeled by $$ D(t)=50 e^{-0.2 t} $$ How many milligrams of the drug remain in the patient's bloodstream after 3 hours?
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