Exponential and logarithmic relations play an important role in mathematics.
Exponential relationships are used in many mathematical models that have an increasing growth rate over time. Logarithmic relations are inverses of exponential relations. For instance, an exponential relation would calculate 3 to the power of 4 and say 81. A logarithmic relation would ask what exponent on 3 makes the result 81? The answer is 4. Here is how both are written.
I'm going to focus on evaluating logs. Let's say you are being asked to compute the following:
Solution: To answer question (a), I need to think about what exponent on 2 would give me an answer of 8. That number is 3. So we write:
To answer question (b), I need to think about what exponent on 5 would give me an answer of 25.
We know that 5 times 5 is 25. So the answer is 2 and is written as:
Let's look at evaluating two logarithms that aren't so obvious.
Solution: (c) You might think that this is impossible. 4 to what exponent gives an answer of 1?
Then you might remember that any non-zero number raised to the power of 0 is also 1, including 4 to the 0. So the answer is 0 and is written as:
(d) Here we know that 6 to the power of 2 is 36, but we want to get 1/36. To create fractions out of
powers we use negative exponents. 6 raised to the power of -2 would become 1 divided by 6 to the power of 2.
Logarithm Laws: There are several laws you will need to know when encountering more than one log
within a problem. Here they are:
Here is an example of how we can use these laws.
The logarithm laws can also be used to convert expressions on the right-side to those on the left-side
as shown below. This is particularly used when dealing with variables rather than numbers.
