So far we have solved exponential equations by

• Finding whole number powers of the base (for example, the solution to is , and the solution to is ).
• Estimation (for example, the solution of is between 2 and 3 because 300 is between 100 and 1,000).
• Using a logarithm and approximating its value on a calculator (for example, the solution of is ).

In the first example, the power of 10 is multiplied by 5, so to find the value of that makes this equation true, each side is divided by 5. From there, the equation is rewritten as a logarithm, giving an exact value for .

In the second example, the expressions on each side of the equation are rewritten as powers of 10: . This means that the exponent on one side and the 3 on the other side must be equal, and leads to an expression to solve where we don't need to use a logarithm.

How do we solve an exponential equation with base , such as ? We can express the solution using the natural logarithm, the logarithm for base . Natural logarithm is written as , or sometimes as . Just like the equation can be rewritten, in logarithmic form, as or , the equation and be rewritten as . Similarly, can be rewritten as .

All this means that we can solve by rewriting the equation as . This says that is the exponent to which base is raised to equal 5.

To estimate the size of , remember that is about 2.7. Because 5 is greater than , this means that is greater than 1. is about , or 7.3. Because 5 is less than , this means that is less than 2. This suggests that is between 1 and 2. Using a calculator we can check that .