Problem 71
Question
Write the exponential equation in logarithmic form. For example, the logarithmic form of \(e^{2}=7.3890 . . .\) is \(\ln 7.3890 . . .=2.\) $$e^{1.3}=3.6692 . . .$$
Step-by-Step Solution
Verified Answer
\(\ln(3.6692...)=1.3\)
1Step 1: Recognize the Exponential
Recognize the given exponential form which is \(e^{1.3}=3.6692 . . .\)
2Step 2: Convert to Logarithmic Form
Rewrite the equation in logarithmic form using the base \(e\), which is the base for the natural logarithm. In this case, it transforms into \(\ln(3.6692...)=1.3\)
Key Concepts
Natural LogarithmExponential EquationBase eConversion Between Forms
Natural Logarithm
The natural logarithm is a special type of logarithm with the base of Euler's number, denoted as \(e\). Unlike common logarithms, which use a base of 10, natural logarithms are based on \(e\), an irrational number approximately equal to 2.71828. The natural logarithm is often abbreviated as \(\ln\). Natural logarithms are deeply rooted in various branches of mathematics due to their natural appearance in sciences like biology and calculus, especially in growth models and compound interest calculations.
The beauty of the natural logarithm lies in how neatly it relates to exponential functions, which is why it's commonly used:
The beauty of the natural logarithm lies in how neatly it relates to exponential functions, which is why it's commonly used:
- Simplifying complex calculus problems.
- Solving exponential equations.
- Transforming complex multiplication into addition.
Exponential Equation
An exponential equation is an equation where the variable appears in the exponent. When dealing with exponential equations, we often encounter a form such as \(e^x = y\). Here, \(e\) is the base and \(x\) is the exponent. Exponential equations are especially useful when modeling situations involving exponential growth or decay, such as population dynamics or radioactive decay.
Solving exponential equations typically involves converting the exponential form into its logarithmic counterpart, which allows us to isolate and solve for the exponent. For example, in the exercise provided, the equation \(e^{1.3} = 3.6692\) is an exponential equation, where "1.3" is the exponent and \(e\) is the base.
Solving exponential equations typically involves converting the exponential form into its logarithmic counterpart, which allows us to isolate and solve for the exponent. For example, in the exercise provided, the equation \(e^{1.3} = 3.6692\) is an exponential equation, where "1.3" is the exponent and \(e\) is the base.
Base e
Euler's number, denoted as \(e\), is one of the most important constants in mathematics. It is approximately 2.71828 and is the base for natural logarithms. This number arises naturally in many contexts, especially where there are continuous growth processes or compounded processes over time. Because of its natural occurrence, it's used extensively in calculations involving growth and decay.
When you see expressions like \(e^x\), it's referring to an exponential function using \(e\) as the base. The choice of base \(e\) is natural and convenient in both mathematical theory and practical applications, such as calculating continuously compounded interest or during the analysis of differential equations.
When you see expressions like \(e^x\), it's referring to an exponential function using \(e\) as the base. The choice of base \(e\) is natural and convenient in both mathematical theory and practical applications, such as calculating continuously compounded interest or during the analysis of differential equations.
Conversion Between Forms
Converting between exponential and logarithmic forms is a key skill in algebra and calculus. The basic principle behind this conversion is understanding that these forms are inverse operations. An exponential equation like \(a^b = c\) can be rewritten in logarithmic form as \(\log_a(c) = b\). This transformation is particularly useful when solving for an unknown exponent in an equation.
In the context of natural logarithms, you often see expressions converted such that \(e^x = y\) becomes \(\ln(y) = x\). This conversion allows us to express systems that are exponential in nature as linear equations, thereby simplifying the process of solving for the unknown.
In the context of natural logarithms, you often see expressions converted such that \(e^x = y\) becomes \(\ln(y) = x\). This conversion allows us to express systems that are exponential in nature as linear equations, thereby simplifying the process of solving for the unknown.
- Logarithmic forms help in solving equations where the unknown is an exponent.
- They assist in simplifying multiplication into addition, aiding in manual calculations.
Other exercises in this chapter
Problem 70
Complete the table to determine the balance \(A\) for \(\$ 12,000\) invested at rate \(r\) for \(t\) years, compounded continuously.. $$\begin{array}{|c|c|c|c|c
View solution Problem 71
Use the properties of logarithms to condense the expression.$$\log _{4} z-\log _{4} y$$.
View solution Problem 71
Solve the exponential equation algebraically. Round your result to three decimal places. Use a graphing utility to verify your answer. $$e^{2 x}-4 e^{x}-5=0$$
View solution Problem 71
Complete the table to determine the balance \(A\) for \(\$ 12,000\) invested at rate \(r\) for \(t\) years, compounded continuously.. $$\begin{array}{|c|c|c|c|c
View solution