## Summary of Integrals, Exponential Functions, and Logarithms

### Essential Concepts

• The earlier treatment of logarithms and exponential functions did not define the functions precisely and formally. This section develops the concepts in a mathematically rigorous way.
• The cornerstone of the development is the definition of the natural logarithm in terms of an integral.
• The function ${e}^{x}$ is then defined as the inverse of the natural logarithm.
• General exponential functions are defined in terms of ${e}^{x},$ and the corresponding inverse functions are general logarithms.
• Familiar properties of logarithms and exponents still hold in this more rigorous context.

## Key Equations

• Natural logarithm function
• $\text{ln}x={\displaystyle\int }_{1}^{x}\frac{1}{t}dt$ Z
• Exponential function$y={e}^{x}$
• $\text{ln}y=\text{ln}({e}^{x})=x$ Z