In this explainer, we will learn how to evaluate logarithms of different bases using laws of logarithms.
You should already know the rules of exponents and how to use them.
Logarithmic functions are the inverse of exponential functions. If the exponential function has an input of , then the output will be .
Now we consider the inverse of this function, which is called a logarithm:
Since this is the inverse of the exponential function, we can input and the output will be .
This means that evaluating is finding the number such that . Alternatively, we could say is such that .
Next, let us look at the definition of a logarithm.
The logarithmic function is the inverse to the exponential function.
means “ raised to the power of equals ,” where
- is the base of the logarithm,
- is the argument,
- is the exponent.
It is important to know how to change an equation from logarithmic form to exponential form, as this will aid us when solving problems involving logarithms. The essential fact is that is equivalent to
There are two special cases of logarithms where the base of the logarithm may not be obvious. These are as follows:
- When a logarithm is written without a base, this means that the logarithm is of base 10; that is,
- When we have a natural logarithm, which is written as , this means that the base of the logarithm is :
Now, we are ready to look at some examples.
Example 1: Calculating a Logarithm
What is the value of ?
What this question means is “what power do we need to raise 2 to in order to obtain 8?” We can rearrange into exponential form to give us which we can solve for .
We know that From the rules of exponents, we know that if , 0, or 1 and then . Therefore, and consequently
Example 2: Finding the Value of a Logarithm
What is the value of ?
To find the value of the logarithm , we are trying to solve the equation
We can convert this equation into exponential form to give us and now we need to solve for . Using the rules of exponents, we know that . Therefore,
Hence, we can see that which gives From this we conclude that
Before we go into some more challenging examples, let us state the rules of logarithms.
The Rules of Logarithms
For any base , , , and real number , the following rules apply:
- Logarithms of 1:
- Logarithms of the base, :
- Logarithms of powers:
- Logarithms of products:
- Logarithms of quotients:
Given that we know how to convert between logarithms and exponentials, it can be useful to see the link between the rules of logarithms and the corresponding rules for exponents.
- The logarithms of 1 rule is equivalent to the zero exponent rule:
- The logarithm of the base rule is equivalent to the exponent rule:
- The logarithms of powers rule does not have a corresponding exponent rule. However, we can see how it is derived algebraically. Using the definition of logarithms, we have that Then, if we raise both sides to the power of , we obtain But, we can also use the definition of logarithms on , which gives us From this, we see that which implies the logarithms of powers rule.
- The logarithms of products rule is equivalent to the product rule for exponents: .
- The logarithms of quotients rule is equivalent to the quotient rule for exponents: .
Using these rules of logarithms, we will consider some more examples.
Example 3: Evaluating a Logarithm
What is the value of ?
First, let us note that we can write . From this, we can use our logarithm of powers rule to obtain
Next, we can use the fact that and substitute this into our logarithm; again, we can use the rule for logarithms of powers to get
Finally, we can use the rule for logarithms of the base to obtain that From this, we get the solution of
Example 4: Using Logarithm Rules to Calculate Logarithms
Calculate , giving your answer to the nearest thousandth.
Using the logarithms of powers rule, we get that
Next, we use the logarithm of products rule to combine these into one logarithm, giving us Now, note that if the logarithm is written without a base, this means the base is 10: In order to get our solution, we just need to calculate this logarithm. We get Finally, we need to round to the nearest thousandth, giving us an answer of
Example 5: Evaluating a Logarithmic Expression
Find the value of .
Using the logarithms of quotients rules, we have that Now, we use the logarithms of powers rule to simplify this:
Next, we us the fact that to obtain
Finally, we use both the logarithms of powers and the logarithms of the base rules to get a solution of
Let us finish by highlighting some key points.
- The logarithmic function is the inverse of the exponential function. Equivalently, evaluating is finding the number such that .
- The two cases where the base of the logarithm is not obvious are and .
- Logarithms of 1: .
- Logarithms of the base: .
- Logarithms of a power: .
- Logarithms of a product: .
- Logarithms of a quotient: .