Problem 59
Question
In Exercises, solve for \(x\) or \(t\). $$ e^{x+1}=4 $$
Step-by-Step Solution
Verified Answer
The solution for the equation \(e^{x+1}=4\) is \(x=\ln{4} - 1\).
1Step 1: Write the equation
Write the given equation, which is \(e^{x+1} = 4\).
2Step 2: Apply the natural logarithm
Apply the natural logarithm on both sides of the equation. This gives: \(\ln{(e^{x+1})} = \ln{4}\). Given that \(\ln{e^a} = a\), this resolves to \(x+1=\ln{4}\).
3Step 3: Solve for x
Solve the resulting equation for \(x\) by subtracting 1 from both sides. This gives the solution: \(x=\ln{4} - 1\).
Key Concepts
Natural LogarithmExponential FunctionsLogarithmic Properties
Natural Logarithm
The natural logarithm, denoted as \( \ln \), is a mathematical function that is the inverse of the exponential function when the base is Euler's number, \( e \). Euler's number is approximately equal to 2.71828, and it is a fundamental constant in mathematics, similar to \( \pi \) in significance.
When dealing with equations like \( e^{x+1} = 4 \), applying the natural logarithm to both sides is an effective method to isolate the variable. Why does this work? The natural logarithm effectively ‘undoes’ the exponential function because if we have \( e^y = x \), then \( \ln x = y \). In our exercise, after applying the natural logarithm, we get to \( x + 1 = \ln 4 \). This happens because \( \ln(e^{x+1}) = x + 1 \), a property that follows from the definition of logarithms.
To interiorize this concept, remember that logarithms, in general, answer the question: 'To what power must we raise the base (in this case, \( e \)) to obtain a certain number?' Therefore, \( \ln 4 \) tells us the power to which we must raise \( e \) to get 4.
When dealing with equations like \( e^{x+1} = 4 \), applying the natural logarithm to both sides is an effective method to isolate the variable. Why does this work? The natural logarithm effectively ‘undoes’ the exponential function because if we have \( e^y = x \), then \( \ln x = y \). In our exercise, after applying the natural logarithm, we get to \( x + 1 = \ln 4 \). This happens because \( \ln(e^{x+1}) = x + 1 \), a property that follows from the definition of logarithms.
To interiorize this concept, remember that logarithms, in general, answer the question: 'To what power must we raise the base (in this case, \( e \)) to obtain a certain number?' Therefore, \( \ln 4 \) tells us the power to which we must raise \( e \) to get 4.
Exponential Functions
Exponential functions are mathematical expressions that involve raising a base number to a variable exponent, typically written in the form \( b^x \), where \( b \) is the base and \( x \) is the exponent. In the context of the given exercise, the exponential function is \( e^{x+1} \), with Euler's number \( e \) as the base.
An important characteristic of exponential functions is that they grow or decay at a rate proportional to their current value. This makes them incredibly useful in modeling phenomena that exhibit rapid changes, such as population growth or radioactive decay.
Understanding how to manipulate exponential functions is crucial. For instance, when you are solving an equation like our exercise, recognizing that you can apply the natural logarithm comes from understanding that the logarithm is the exponential function’s inverse. This opens the door to simplifying the equation and ultimately, solving for the variable.
An important characteristic of exponential functions is that they grow or decay at a rate proportional to their current value. This makes them incredibly useful in modeling phenomena that exhibit rapid changes, such as population growth or radioactive decay.
Understanding how to manipulate exponential functions is crucial. For instance, when you are solving an equation like our exercise, recognizing that you can apply the natural logarithm comes from understanding that the logarithm is the exponential function’s inverse. This opens the door to simplifying the equation and ultimately, solving for the variable.
Logarithmic Properties
Logarithmic properties are rules that make working with logarithms easier and are deeply interconnected with the properties of exponents, given that logarithms are the inverse of exponentiation. Some of these properties include the product rule, the quotient rule, and the power rule.
Here's how they relate:
Here's how they relate:
- The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors, \( \ln(ab) = \ln a + \ln b \).
- The quotient rule explains that the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator, \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \).
- The power rule states that the logarithm of a number raised to an exponent is equal to the exponent times the logarithm of the base, \( \ln(a^b) = b\cdot\ln a \).
Other exercises in this chapter
Problem 58
In Exercises, use implicit differentiation to find an equation of the tangent line to the graph at the given point. $$ y^{2}+\ln (x y)=2, \quad(e, 1) $$
View solution Problem 58
In Exercises, solve for \(x\) or \(t\). $$ 2 \ln 4 x=7 $$
View solution Problem 59
In Exercises, find the second derivative of the function. $$ f(x)=x \ln \sqrt{x}+2 x $$
View solution Problem 60
In Exercises, solve for \(x\) or \(t\). $$ e^{-0.5 x}=0.075 $$
View solution