### 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 [latex]{e}^{x}[/latex] is then defined as the inverse of the natural logarithm.
- General exponential functions are defined in terms of [latex]{e}^{x},[/latex] 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**- [latex]\text{ln}x={\displaystyle\int }_{1}^{x}\frac{1}{t}dt[/latex] Z
**Exponential function**[latex]y={e}^{x}[/latex]- [latex]\text{ln}y=\text{ln}({e}^{x})=x[/latex] Z