PROPERTIES OF LOGARITHMS

Property 1: because .

Example 1: In the equation , the base is 14 and the exponent is 0. Remember that a logarithm is an exponent, and the corresponding logarithmic equation is where the 0 is the exponent.

Example 2: In the equation , the base is and the exponent is 0. Remember that a logarithm is an exponent, and the corresponding logarithmic equation is .

Example 3: Use the exponential equation to write a logarithmic equation. The base x is greater than 0 and the exponent is 0. The corresponding logarithmic equation is .

Property 2: because .

Example 4: In the equation , the base is 3, the exponent is 1, and the answer is 3. Remember that a logarithm is an exponent, and the corresponding logarithmic equation is .

Example 5: In the equation , the base is 87, the exponent is 1, and the answer is 87. Remember that a logarithm is an exponent, and the corresponding logarithmic equation is .

Example 6: Use the exponential equation to write a logarithmic equation. If the base p is greater than 0, then .

Property 3: because .

Example 7: Since you know that , you can write the logarithmic equation with base 3 as .

Example 8: Since you know that , you can write the logarithmic equation with base 13 as .

Example 9: Use the exponential equation to write a logarithmic equation with base 4. You can convert the exponential equation

to the logarithmic equation . Since the 16 can be written as

, the equation can be written .

The above rules are the same for all positive bases. The most common bases are the base 10 and the base e. Logarithms with a base 10 are called common logarithms, and logarithms with a base e are natural logarithms. On your calculator, the base 10 logarithm is noted by log, and the base e logarithm is noted by ln.

There are an infinite number of bases and only a few buttons on your calculator. You can convert a logarithm with a base that is not 10 or e to an equivalent logarithm with base 10 or e. If you are interested in a discussion on how to change the bases of a logarithm, click on Change of Base.

For a discussion of the relationship between the graphs of logarithmic functions and exponential functions, click on graphs.

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