Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Exponential & Logarithmic Equations

Calculus

0. Functions

Exponential & Logarithmic Equations

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2:15 minutes

Problem 1.55

Textbook Question

Solving equations Solve the following equations.

ln x= -1

Verified step by step guidance

Step 1: Understand the equation. The equation given is \( \ln x = -1 \). This means we are looking for a value of \( x \) such that the natural logarithm of \( x \) equals \(-1\).

Step 2: Recall the definition of the natural logarithm. The natural logarithm \( \ln x \) is the power to which \( e \) (Euler's number, approximately 2.718) must be raised to obtain \( x \).

Step 3: Convert the logarithmic equation to an exponential form. The equation \( \ln x = -1 \) can be rewritten as \( x = e^{-1} \).

Step 4: Simplify the expression. The expression \( e^{-1} \) can be simplified to \( \frac{1}{e} \).

Step 5: Conclude the solution. The value of \( x \) that satisfies the equation \( \ln x = -1 \) is \( \frac{1}{e} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is the inverse function of the exponential function, meaning that if y = ln(x), then x = e^y. Understanding the properties of natural logarithms is essential for solving equations involving ln.

Exponential Function

The exponential function is a mathematical function denoted as e^x, which describes growth or decay processes. It is crucial for solving logarithmic equations, as converting a logarithmic equation to its exponential form allows for easier manipulation. For example, if ln(x) = -1, it can be rewritten as x = e^(-1).

Solving Logarithmic Equations

Solving logarithmic equations involves isolating the variable by converting the logarithmic expression into its exponential form. This process often requires understanding the properties of logarithms, such as the product, quotient, and power rules. In the case of ln(x) = -1, applying the exponential function helps find the value of x.