# Exponential and logarithmic equation

The following rules are true for logarithms We justify the first one using a numerical example Example 1 Example 2 - Solve the equation On the other hand, the equation of Example 2 can also be solved as follows So So we have changed the 3-base logarithm to a 10-base logarithm. In general, for changing the base number of logarithm Example 3 - Solve the equation We change both sides to have base number 4 Exponential function 4x is strictly increasing Example 4 - Solve the equation Domain of definition We solve the equation The solution is valid

Example 5 - Solve the equation The equation is defined when  We solve the equation The solution is valid

Example 6 - Solve the inequality The inequality is defined when x > 0

We say the number 3 with the 5-base logarithm The base number 5 > 1, i.e. the function log5x , is strictly increasing. So the direction of the inequality is maintained. Example 7

Which of the following numbers is greater?

Usually, calculators can't compute numbers this big We use the help of the 2-base logarithm And with a calculator, the latter is Since log2 is a strictly increasing function, the larger of the above numbers is the one with the larger 2-base logarithm, i.e. the number Turn on the subtitles if needed