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Chapter 3: Problem 53

Write the exponential equation in logarithmic form. $$e^{2}=7.3890 \ldots$$

Short Answer

Expert verified
The logarithmic form of \(e^2 = 7.3890\ldots\) is \(ln(7.3890\ldots) = 2\).

Step by step solution

01

Identify the Base, Exponent and Result

In the exponential equation \(e^2 = 7.3890\ldots\), \(e\) is the base, \(2\) is the exponent and \(7.3890\ldots\) is the result.
02

Convert to logarithmic form

Rewrite the equation in logarithmic form according to the principle \(log_b x = y\). In this case, the base \(b\) is \(e\), \(x\) is \(7.3890\ldots\), and \(y\) is \(2\). So, we have \(log_e 7.3890\ldots = 2\). Since \(e\) is the natural base, it is commonly written as \(ln\) instead of \(log_e\).

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Key Concepts

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

Exponential Equations
An exponential equation involves a variable as an exponent. In such equations, you see expressions like \(a^b = c\), where \(a\) is the base, \(b\) is the exponent, and \(c\) is the result. Exponential equations are fundamental in mathematics, especially when dealing with growth, decay, or complex financial calculations.

Solving exponential equations often requires transforming them into a different form. This might include either converting into a logarithmic form or isolating the variable to reveal insights about its value.
  • Base: The number being raised to a power. In \(e^2 = 7.3890\ldots\), the base is \(e\).
  • Exponent: The power the base is raised to, such as \(2\) in \(e^2\).
  • Result: The outcome of raising the base to the exponent, like \(7.3890\ldots\) in our example.
Understanding how to navigate these components helps in managing exponential equations.
Natural Logarithm
The natural logarithm is a specific type of logarithm with the base \(e\), where \(e\approx 2.71828\). It's represented as \(\ln\) when expressed without a base number. Natural logarithms are widely used in exponential growth processes like population growth or radioactive decay.

Using natural logarithms transforms exponential equations into a more solvable equation:
  • The formula \(e^y = x\) converts into \(\ln(x) = y\).
  • For example, the exponential equation \(e^2 = 7.3890\ldots\) converts to \(\ln(7.3890\ldots) = 2\).
This conversion is vital in calculus and other higher mathematics, allowing us to solve equations with exponential terms more efficiently.
Base e
The constant \(e\) is an irrational number approximately equal to 2.71828, and it's the base of natural logarithms. Its special properties make it pivotal in mathematics, especially concerning exponential growth and decay.

Using \(e\) as a base in exponential functions offers advantages:
  • Its derivative, when used in functions like \(e^x\), equals \(e^x\), simplifying calculus calculations.
  • It models natural phenomena accurately, making \(e\) central in fields like physics, biology, and economics.
The base \(e\) turns out to be the limit as \(n\) approaches infinity for the expression \(\left(1 + \frac{1}{n}\right)^n\), capturing the essence of continuous growth.

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Most popular questions from this chapter

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$10-4 \ln (x-2)=0$$

Use the properties of logarithms to expand the expression as a sum, difference, and or constant multiple of logarithms. (Assume all variables are positive.) $$\ln x y z^{2}$$

Use the properties of logarithms to expand the expression as a sum, difference, and or constant multiple of logarithms. (Assume all variables are positive.) $$\log _{5} \frac{x^{2}}{y^{2} z^{3}}$$

The effective yield of an investment plan is the percent increase in the balance after 1 year. Find the effective yield for each investment plan. Which investment plan has the greatest effective yield? Which investment plan will have the highest balance after 5 years? (a) \(7 \%\) annual interest rate, compounded annually (b) \(7 \%\) annual interest rate, compounded continuously (c) \(7 \%\) annual interest rate, compounded quarterly (d) \(7.25 \%\) annual interest rate, compounded quarterly

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$6 e^{1-x}=25$$
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