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 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

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