In this explainer, we will learn how to evaluate logarithms of different bases using laws of logarithms.
First, recall that a logarithm is the inverse function of an exponential function. As such, we can write the same expression either as an exponential equation or as a logarithmic equation by rearranging the equation as follows.
Definition: Logarithms
For an exponential equation in the form , where , this can be written as the logarithmic equation where is the base of the logarithm, is the argument, and is the exponent.
Recall that we can use the definition above to help us evaluate logarithms. For example, if we wanted to evaluate this would be the same as asking, β4 to the power of what is 16?β.
Or, if we set , equal to , then so
As we know, 16 is ; therefore,
Hence,
We will explore a similar question in our first example.
Example 1: Evaluating a Logarithm
What is the value of ?
Answer
To find the value of , we can use the definition of a logarithm to rewrite this as an exponential equation. We know that where .
By setting equal to , we can rewrite this as
Therefore, we need to find 2 to the power of what number is equal to 8. This gives us 3, so for
So,
Therefore, the answer is 3.
When evaluating logarithms with noninteger bases or arguments, it is helpful to use the laws of exponents for fractional indices and negative indices, as shown in the property below.
Property: Laws of Exponents for Fractional and Negative Indices
- ,
- ,
In the following example, we will consider how to use the law of exponents for fractional indices to help us evaluate a logarithm.
Example 2: Evaluating Logarithms
What is the value of ?
Answer
To find the value of , it is helpful to write the argument, , as a power of the base, 2.
Using the law of exponents for negative indices, where , we can write as a power of 2 as follows:
This then gives us
We can use the definition of a logarithm to rewrite this as an exponential equation. We know that where . By setting , we then get
By comparing exponents in , we can see that , which is the value of .
Often, we can use a calculator to evaluate the value of a logarithm. We can do this using the logarithm function and inputting the argument and base. In the next example, we will use a calculator to evaluate the value of the logarithm.
Example 3: Evaluating Logarithms Using a Calculator
Find the value of correct to 4 decimal places.
Answer
To find the value of correct to 4 decimal places, we will need to use a scientific calculator.
When evaluating this logarithm, we are asking, β4 to the power of what number gives us 22.08?β It is helpful to estimate the answer first before using the calculator.
Thinking about the powers of 4, we know that the closest powers to 22.08 are , which is 16, and , which is 64. This means that the power will be between 2 and 3, meaning the value of will be between 2 and 3.
To use a scientific calculator, we do the following steps.
First, we need to locate the button for inputting a logarithm. On your scientific calculator or graphing display calculator, you need to find the button that has the symbol shown below.
Second, once we have pressed the log button, we will then be able to input 4 as the base and 22.08 as the argument, as shown below.
Lastly, when we press the equals button, we get the value of the logarithm correct to 10 significant figures, which is 2.232334134, as shown below.
Once we have obtained the value from the calculator, we can then round this to 4 decimal places, giving us 2.2323.
Therefore, the value of is 2.2323 correct to 4 decimal places.
In addition to evaluating logarithms, given the base and the argument, we can also find unknown bases or arguments in a logarithm by rewriting it in its exponent form. In the next example, we will demonstrate how to do this for an unknown in the argument.
Example 4: Finding Unknowns in a Single Logarithm
If , what is the value of ?
Answer
To find the value of in , we can first write the logarithm in its exponent form. We know that where .
Therefore,
Simplifying and solving , we get
Therefore, if , the value of is 737.
We can then check the answer by substituting this back into the expression. This gives us
As we know 9 to the power of 3 is 729, we can see that this is the correct answer.
Sometimes, when working with logarithms, we use what is called the βcommon logarithm,β which has base 10. We use a special notation for this, as shown in the definition below.
Definition: Notation for Logarithms with Base 10
For a logarithm with base 10, we omit the base and only write log. That is, which is called the βcommon logarithm.β
In our last example, we will evaluate a logarithm with base 10 and then simplify this.
Example 5: Evaluating a Logarithm with Base 10
Given that , find the value of .
Answer
To find the value of when , we will start by substituting into the expression. Doing so gives us
To simplify the first term inside the logarithm, we need to multiply the numerator and denominator of the fraction by the conjugate of , which is . This then gives us
Simplifying the terms inside the logarithm, we get
As is , we know that this is 1, since 10 to the power of 1 is 10.
Therefore, the value of when is 1.
In this explainer, we have learned how to evaluate logarithms by rewriting them as exponents and, where necessary, using the laws of exponents. Letβs recap the key points.
Key Points
- We can rewrite logarithms in exponential form in order to help evaluate them: where .
- When working with fractional or negative indices, we can use the laws of exponents to help us evaluate logarithms.
- We can use a scientific calculator to evaluate logarithms where needed.
- When finding unknowns in single logarithms, we can first rewrite them in their exponent form and then solve for the unknown.
