1. Logarithmic function operation: addition, subtraction, multiplication and division of logarithmic function, and reciprocal operation of exponent and logarithm are all important applications of logarithm.

2. Study on the properties of exponential function: Study the properties of exponential function through the properties of logarithmic function, for example, deduce the commonly used exponential formula by using the formula of changing the logarithm base and the arithmetic of logarithm (for example, the x power of e equals the x times lnx of e).

3. Solution of exponential equation: The solution of exponential equation often needs the concept and method of logarithm. For example, when solving the equation A x = B, solving the logarithmic value of A can be transformed into solving the logarithmic value of B. ..

4. Interoperability between exponent and logarithm: In scientific calculation and data analysis, it is often necessary to convert exponential data into logarithmic data or logarithmic data into exponential data for further processing and analysis.

5. Solution of logarithmic inequality: Logarithmic inequality is a special inequality. By moving the logarithmic sign aside, inequality can be transformed into other types of problems, which makes it easier to solve.

6. Application in statistics: Logarithmic functions are often used in normal distribution, T distribution and other issues in statistics. These distributions can be converted into standard normal distribution by logarithmic transformation, which is convenient for analysis and comparison.

7. Application in computer science: In computer science, logarithmic functions are often used in algorithm analysis and data structure design, such as method of bisection algorithm and quick sorting.

These are just some common applications of logarithm in mathematics. In fact, logarithm has important applications in many fields of mathematics and natural science.