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

Properties of Logarithms

Calculus

0. Functions

Properties of Logarithms

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1:29 minutes

Problem 77e

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

$2 = \ln 2^{e}$

Verified step by step guidance

Step 1: Start by analyzing the given equation: $ 2 = \ln(2^e) $.

Step 2: Recall the logarithmic identity $ \ln(a^b) = b \cdot \ln(a) $.

Step 3: Apply the identity to the right side of the equation: $ \ln(2^e) = e \cdot \ln(2) $.

Step 4: Substitute back into the equation: $ 2 = e \cdot \ln(2) $.

Step 5: Solve for $ e $ by dividing both sides by $ \ln(2) $: $ e = \frac{2}{\ln(2)} $.

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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 a fundamental concept in calculus, particularly in relation to exponential functions. The natural logarithm has properties that make it useful for solving equations involving exponential growth or decay, and it is often used in integration and differentiation.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where a and b are constants, and e is the base of the natural logarithm. These functions exhibit rapid growth or decay and are characterized by their constant percentage rate of change. Understanding exponential functions is crucial for analyzing growth models, compound interest, and natural phenomena.

Equality of Functions

To determine if two expressions are equal, one must evaluate both sides under the same conditions. In calculus, this often involves substituting values or simplifying expressions. For the statement 2 = ln(2^e), one must understand how to manipulate logarithmic identities and evaluate the left and right sides to verify their equality or find a counterexample.