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Chapter 4: Problem 3

If \(e^{x}=5,\) write \(x\) in terms of the natural logarithm.

Short Answer

Expert verified
x = \ln(5)

Step by step solution

01

Understand the given equation

The equation provided is \(e^{x}=5\). Here, you need to solve for \x\ using the natural logarithm function.
02

Apply natural logarithm to both sides

To isolate \x\, apply the natural logarithm, denoted as \ln\, to both sides of the equation: \[\ln(e^{x}) = \ln(5)\].
03

Use logarithm properties

Use the logarithm property \(\ln(e^{x}) = x \), which simplifies the left-hand side of the equation: \[ x = \ln(5) \].
04

Write final expression for x

The value of \x\ in terms of the natural logarithm is now isolated. Thus, \[ x = \ln(5) \].

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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 is an equation in which a variable appears in the exponent. These types of equations are in the form of \(a^{b} = c \) where you often solve for \(b\).Understanding exponential equations is crucial because they model many real-world phenomena like population growth or radioactive decay.
To solve these kinds of equations, we make use of logarithms. This leads us to very useful techniques, especially involving the natural logarithm (ln).
logarithm properties
Logarithms have several helpful properties that often simplify equations and make them easier to solve. Three commonly used properties are:
  • Product Property: \ln(ab) = \ln(a) + \ln(b)\
  • Quotient Property: \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\
  • Power Property: \ln(a^b) = b \ln(a)\
Applying these properties makes solving logarithmic equations a step-by-step process rather than a guessing game.
In the provided example, we used the power property to solve the equation \( \ln(e^x) = x \). This property states that taking the logarithm of an exponential function with base \(e\), the natural logarithm and base cancel each other out.
solving equations
Solving equations often involves isolating the variable of interest. Let's look at the given example again: \(e^{x} = 5\). Here's the step-by-step breakdown:
First, we start with the equation \(e^{x} = 5\). To isolate \(x\), take the natural logarithm of both sides: \ \ln(e^{x}) = \ln(5) \
Using the logarithm property \ \ln(e^{x}) = x \, we simplify the left-hand side to get \ x = \ln(5) \
Thus, the solution to the equation \ e^{x} = 5 \ is \ x = \ln(5) \.Practicing these steps with different equations can significantly improve your problem-solving skills.

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Most popular questions from this chapter

Differentiate the following functions. $$y=\frac{x+1}{e^{x}}$$

Solve the given equation for \(x .\) $$\ln x-\ln x^{2}+\ln 3=0$$

Solve the given equation for \(x .\) $$2(\ln x)^{2}+\ln x-1=0$$

Find the values of \(x\) at which the function has a possible relative maximum or minimum point. (Recall that \(e^{x}\) is positive for all \(x .\) ) Use the second derivative to determine the nature of the function at these points. $$f(x)=\frac{4 x-1}{e^{x / 2}}$$

Find the slope-point form of the equation of the tangent line to the graph of \(e^{x}\) at the point \(\left(a, e^{a}\right).\)
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