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Chapter 3: Problem 53

Writing a Natural Logarithmic Equation In Exercises \(53-56,\) write the exponential equation in logarithmic form. $$e^{1 / 2}=1.6487 \ldots$$

Short Answer

Expert verified
The log form of the given equation is \(\ln 1.6487 = 1/2\).

Step by step solution

01

Identify the Base, Exponent, and Result

In the given equation \(e^{1/2} = 1.6487 \ldots\), the base is e, the exponent is \(1/2\), and the result is approximately \(1.6487\). We use these three elements to write the logarithmic equivalent.
02

Conversion to Logarithmic Form

The logarithmic form of an equation \(b^x = y\) is \(\log_b y = x\). Replace b with e, x with \(1/2\), and y with \(1.6487\) to get the logarithmic form.
03

Write the logarithm in Natural Logarithmic Form

We typically write logarithms with base e as natural logarithms, denoted by ln. Hence, write \(\log_e 1.6487 = 1/2\) as \(\ln 1.6487 = 1/2\).

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Key Concepts

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

Exponentiation
Exponentiation is a fundamental concept in mathematics where a number, known as the base, is multiplied by itself a certain number of times, known as the exponent. It is represented in the form \( b^x \), where \( b \) is the base and \( x \) is the exponent. This operation is key in understanding growth processes and scaling factors across various fields.
  • Base: The number being multiplied.
  • Exponent: Indicates how many times the base is used as a factor.
  • Result: The final value obtained after exponentiation.
Let's consider the expression \( e^{1/2} = 1.6487 \ldots \). Here, \( e \) serves as the base, while \( \frac{1}{2} \) is the exponent. The result is approximately 1.6487. Exponentiation is used in many natural phenomena, such as population growth, radioactive decay, and compound interest.
Logarithmic Form
The logarithmic form is a way to express exponential equations as logarithmic equations, which often simplifies complex multiplicative processes. If you have an exponential equation \( b^x = y \), it can be rewritten in logarithmic form as \( \log_b y = x \).
  • Original form: exponential \( \Rightarrow \) logarithmic
  • Helps in solving equations: especially useful for finding exponents.
For the equation \( e^{1/2} = 1.6487 \), converting to logarithmic form involves identifying the base, which is \( e \), the exponent \( \frac{1}{2} \), and the result which is 1.6487. The logarithmic form becomes \( \log_e 1.6487 = 1/2 \). This transformation allows us to solve for unknown exponents more effectively.
Base e
The base \( e \) is a mathematical constant approximately equal to 2.71828. It is the foundation for natural logarithms and arises frequently in calculations involving growth and decay. Known as Euler's number, \( e \) is crucial in fields like calculus, statistics, and financial mathematics.
  • Value: Approximately 2.71828
  • Appearance: Natural growth and decay processes
  • Properties: Logarithms with base \( e \) are called natural logarithms.
When working with natural logarithms, the base \( e \) is implied, particularly in natural sciences and engineering, where continuous growth processes are modeled. For the conversion example \( e^{1/2} = 1.6487 \), \( e \) serves as the base in both the exponential and its corresponding logarithmic forms.
Exponential Equations
Exponential equations feature prominently in mathematics to describe relationships involving exponential growth or decay. These equations take the form \( b^x = y \), where solving for \( x \) requires converting the equation into a logarithmic form.
  • Growth: Used in populations, economies, etc.
  • Decay: Used in radioactive processes, cooling, etc.
  • Solution: Usually involves logarithms for handling the exponents effectively.
In our example, the exponential equation \( e^{1/2} = 1.6487 \) demonstrates a simple exponential relationship. By understanding the structure of exponential equations, we can effectively analyze and solve real-world problems that exhibit exponential behavior. Changing these into logarithmic form is often the first step to finding unknown values within the equation.

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Most popular questions from this chapter

In Exercises \(103-106,\) use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph the ratio. $$f(x)=\log _{1 / 2} x$$

Find the domain, \(x\) -intercept, and vertical asymptote of the logarithmic function and sketch its graph. $f(x)=\log _{4} x$$

Use a graphing utility to graph the functions \(y_{1}=\ln x-\ln (x-3)\) and \(y_{2}=\ln \frac{x}{x-3}\) in the same viewing window. Does the graphing utility show the functions with the same domain? If so, should it? Explain your reasoning.

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$4 \log (x-6)=11$$

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$3-\ln x=0$$
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