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Chapter 3: Problem 9

Change each equation to its exponential form. $$\ln 1=0$$

Short Answer

Expert verified
The exponential form of the given equation \(\ln 1 = 0\) is \(e^0 = 1\).

Step by step solution

01

Identify the components of the logarithmic equation

In the logarithmic form \(\ln b = a\), 'e' is the base (which is implicit in the natural log symbol), \(b\) is the argument, and \(a\) is the result of the operation.
02

Change to exponential form

Use the fundamental relationship between logarithms and exponentials. The base 'b' of the exponential form is the base of the logarithm, the exponent 'a' is the result of the logarithm, and 'c' is the argument of the logarithm. Thus, the equivalent exponential form of \(\ln b = a\) is \(e^a = b\).
03

Apply the transformation for the given equation

Substitute the given values from \(\ln 1 = 0\) into the exponential form. Hence, the corresponding exponential form is \(e^0 = 1\).

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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 is a special type of logarithm where the base is a unique number denoted by 'e'. It is commonly used in mathematics because it features prominently in calculus and complex calculations.
Denoted by the symbol \(\ln\), it specifically refers to a logarithm where the base is the mathematical constant \(e\), which is roughly equal to 2.71828.
Unlike the common logarithms, which use a base of 10, natural logarithms are used to simplify expressions involving powers and exponential functions.
  • Natural logarithms are useful in solving equations where the variables are in the exponents.
  • They connect exponential growth and decay processes, which makes them crucial in fields like finance, physics, and biology.
Understanding natural logarithms helps to grasp more complex mathematical concepts and solve exponential and logarithmic equations efficiently.
Exponential Form
The exponential form of an equation is a way of expressing numbers as powers or exponents of a base number, typically using the constant \(e\).
When we talk about the exponential form in relation to logarithms, it involves expressing the logarithmic equation in a way that shows the power to which the base must be raised.
For a natural logarithm, the equation \(\ln b = a\) is transformed to its exponential form as \(e^a = b\). This signifies that the base, \(e\), raised to the power of \(a\), equals \(b\).
  • The exponential form is essential for solving logarithmic equations because it directly relates to the base and the result of the logarithmic operation.
  • It provides a clear method to convert problems into a format that is easier to manage and calculate.
Being comfortable with exponential forms is crucial for complex problem-solving in both pure and applied mathematics.
Logarithmic Form
The logarithmic form of an equation is a way to express the power to which a base must be raised to produce a given number.
In the context of natural logarithms, this means turning an exponential relationship into a logarithmic one.
For example, if an exponential equation is \(e^a = b\), its logarithmic form would be \(\ln b = a\). Here, \(\ln\) denotes that the base of the logarithm is 'e'.
  • Logarithmic forms are incredibly useful because they simplify the multiplication and division of large numbers into manageable addition and subtraction operations.
  • They are especially handy in calculus for finding derivatives and integrals of exponential functions.
By converting to logarithmic form, you can often make complex equations more approachable and easier to solve.
Base e
The base \(e\) is a mathematical constant approximately equal to 2.71828, and it is the foundation of natural logarithms.
It is often called Euler's number after the mathematician Leonhard Euler, who conceptualized its importance in mathematics.
The constant \(e\) arises naturally in situations involving continuous growth or decay, making it highly significant in many scientific fields including biology, economics, and physics.
  • It is the limit of \((1 + \frac{1}{n})^n\) as \(n\) approaches infinity, which also defines the rate of growth continuously compounded.
  • The base \(e\) is unique in its natural appearance in various mathematical contexts like calculating compound interest continuously or modeling population growth.
Understanding 'e' is critical for mastering concepts in calculus, particularly those involving exponential and logarithmic functions.

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Most popular questions from this chapter

Evaluate the exponential function for the given \(x\) -values. $$j(x)=\left(\frac{1}{2}\right)^{x} ; x=-2 \text { and } x=4$$

Use a calculator to evaluate the exponential function for the given \(x\) -value. Round to the nearest hundredth. $$g(x)=e^{x}, x=2.2$$

Data from Forrester Research suggest that the number of broadband [cable and digital subscriber line (DSL) \(]\) connections to the Internet can be modeled by \(f(x)=1.353(1.9025)^{x},\) where \(x\) is the number of years after January \(1,1998,\) and \(f(x)\) is the number of connections in millions. a. How many broadband Internet connections, to the nearest million, does this model predict will exist on January \(1,2005 ?\) b. According to the model, in what year will the number of broadband connections first reach 300 million? [Hint: Use the intersect feature of a graphing utility to determine the \(x\) -coordinate of the point of intersection of the graphs of \(f(x)\) and \(y=300.1\)

A medical care package is air lifted and dropped to a disaster area. During the free-fall portion of the drop, the time, in seconds, required for the package to obtain a velocity of \(v\) feet per second is given by the function $$t=2.43 \ln \frac{150+v}{150-v}, \quad 0 \leq v<150$$ a. Determine the velocity of the package 5 seconds after it is dropped. Round to the nearest foot per second. b. Determine the vertical asymptote of the function. c. Write a sentence that explains the meaning of the vertical asymptote in the context of this application.

The percent \(I(x)\) of the original intensity of light striking the surface of a lake that is available \(x\) feet below the surface of the lake is given by \(I(x)=100 e^{-0.95 x}\) a. What percentage of the light, to the nearest tenth of a percent, is available 2 feet below the surface of the lake? b. At what depth, to the nearest hundredth of a foot, is the intensity of the light one-half the intensity at the surface?
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