The key to understanding logarithms is through their relationship with exponential functions. Since is the inverse function to , many of the properties of exponential functions can be translated into properties of logarithms. In this section, we’ll try to discover these and find several other interesting properties of logarithms along the way.
We highlight several important principles from our previous discussion of inverse functions:
- A function has an inverse function if and only if there exists a function that undoes the work of : that is, there is some function for which for each in the domain of , and for each in the range of . We call the inverse of , and write .
- When has an inverse, we know that writing “” and “” are two different perspectives on the same statement.
Inverse Property of Logarithms
An important fact to recall is that the range of the function is , the set of all positive real numbers. This means that any positive real number can be written as the output of the exponential function with base . Let’s fix and try to write the number 17 as an output of the function . If 17 is an output of , then for some real number . Taking of both sides of this equation, we find that .
Now we use the most important property of logarithms: the logarithms and exponential of the same base are inverses. With our base being set to 10, this tells us that . It is important to remember that even though our notation for the exponential function writes its input as an exponent, and not by wrapping it in parenthesis, is the input to the exponential function in .
Returning to our original quest to write 17 as an output of the exponential with base 10, we use the inverse property of logarithms to say that , and therefore,
Another way to see this is by using the fact that the function is the inverse of .
There was nothing special about 10 and 17 in what we just showed, so this allows us to arrive at a very general way to write positive real numbers as exponentials.
Another way to understand this is to remember the definition of the logarithm. is precisely the power to which you have to raise in order to obtain .
Finally, this can also be viewed as a statement about inverse functions. If , then . In this setup, the statement becomes .
Product Property of Logarithms
You might think that the method in the previous section of writing positive real numbers as exponentials unnecessarily complicates things, but we can use it to adapt properties of exponents into properties of logarithms.
Recall that multiplying exponential expressions of the same base results in another exponential expression whose exponent is the sum of the two original exponents: in symbols, for any real numbers and .
Let’s see if we can use this fact, again restricting our attention to . Since 2 and 3 are positive real numbers, we can write and . Then,
Notice again how we used the fact that the logarithm and exponential with base 10 are inverses! There’s nothing special about 2 and 3, so for any positive real numbers and , . Even more, there’s nothing special about base 10, allowing us to come up with a general rule.
Quotient Property of Logarithms
Now that we’ve dealt with multiplication, it makes sense to deal with division. If and are positive real numbers, we can think about the quotient as a product: . What’s more, we can write as a power of : . Using the product property of logarithms from the previous section, we can conclude that .
It would be really nice if there was a nice relationship between and . Indeed, there is! Using the definition of the logarithm, is the power to which you have to raise to obtain , but to obtain , we can use the negative power. As an example, note that , but . In general,
Combining this with our previous work, we obtain the following quotient property of logarithms.
Power Property of Logarithms
Something else you might remember about exponents is that repeated exponentiation is the same thing as multiplying exponents. For example, (check this yourself!). In words, this says that raising 7 to the 3rd power, then raising that result to the 2nd power is the same as raising 7 to the th power. Since , . So in the language of logarithms, the above says that .
In general, for all real numbers , , and .
Let’s see if this fact has any consequences for logarithms! Recall that for positive and , is the power to which we need to raise in order to obtain . However, another way to obtain is to raise to the power (yielding ) and then raise that result to the power . Since repeated exponentiation is the same thing as multiplying exponents, this amounts to raising to the power . In symbols, we’ve shown that
In essence, taking the logarithm of a power of is the same thing as multiplying the logarithm of by the power. An intuitive way to think about this property is in the context of the product property from above. Since logarithms “turn multiplication into addition” and exponentiation is repeated multiplication, logarithms should “turn exponentiation into repeated addition”, that is, multiplication. As an example, notice that
The above calculation uses the product property to arrive at the same conclusion as the power property.Change-of-Base Formula
One important thing to recognize is that logarithms can have any positive number (except 1) as their base. Sometimes, when doing calculations, it may be preferable to use one base over another. The good news is that any logarithm can be computed using this preferred base.
As an example, consider the quantity . Many calculators are unable to directly calculate logarithms with a base other than or 10, so let’s convert this into a natural logarithm (logarithm with base ). Rewriting 7 as using the inverse property of logarithms, we see that . Now, using the power property of logarithms, we see that . This gives us the equality , so dividing both sides by , . If you have an aversion to and a fondness for , then this allows you to calculate instead of .
Of course, there’s nothing special about 3, 7, and the natural logarithm. In general, we have the following formula.
In words, this formula says that instead of using to calculate , we can make two calculations with and divide, which will yield the same result.
Logarithm Properties in Action
In the previous two examples, we illustrated a trade-off that occurs. When we combine logarithms into one, we often find that their arguments become messy. However, in order to simplify the arguments of logarithms, we need to separate them out into sums and differences of logarithms. We can have simple arguments, but multiple logs, or we can have one log, but complex arguments.