Review of Inverses
Before we talk about logarithms, let's do a quick review of inverses. Inverse operations are operations that "undo" each other. For example, addition and subtraction are inverses. If you add 5, you can easily undo that by subtracting 5. Multiplication and division are also inverses. If you multiply a number by 3, you can undo that by dividing by 3.
So what's a logarithm? It's an operation that will undo an exponent. Logarithms are the inverses of exponential functions. Why the funny name? You'll have to ask John Napier - he came up with the name back in the 1600s. We use logarithms to go backwards and solve an exponential equation. They help us figure out this type of problem:
We can use a logarithm to figure out what the exponent must be for this to come out to 125. In the exponent above, the base is 5. That means 5 times itself how many times will equal 125? A logarithm will tell us the answer. When we write logarithms, we use the same base that was used in the exponential equation. The base in this example is 5, so we'll have a logarithm with a base of 5. This is written as:
When we write logarithms, we abbreviate them and write log instead. The small number written as a subscript after the log indicates what the base is. You would read this as "log base 5 of 125". It means 5 to what power will equal 125?
Do you know the answer? 5 times itself how many times will equal 125? We know the answer isn't 2 because 5 to the second power is equal to 25. 5 to the 3rd power is 5x5x5 = 125. This means the answer is 3.
Changing Between Exponential and Logarithmic Form
You can rewrite an exponential equation as a logarithm and vice versa. Remember, both forms always use the same base. The base of the logarithm (the little number written after the word log) will be the same as the base of the exponent (the number that's being multiplied by itself).
Rewrite in logarithmic form
First, identify the base. 2 is being raised to the 8th power - this means that 2 is the base. The logarithm will have the same base, so we know to write a little 2 as a subscript after the word log.
Remember, a logarithm tells you what the exponent is. In the exponential equation above, the exponent is 8. This means the logarithm is equal to 8. The number after log base 8 is what you want the power to come out to - we want it to come out to 256.
That's it! It's now in logarithmic form.
Rewrite in exponential form.
First, identify the base. The exponential and logarithmic equations are both going to have the same base. The equation is written in logarithmic form right now, so the base is the little number right after the word log - the 3. This means our equation in exponential form will also have a base of 3.
The next step is to figure out what the exponent is. Remember, the logarithm tells you what the exponent is. This logarithm is equal to 6 - this means the exponent is 6.
Common Logarithms - Base 10
A logarithm can have any positive number as the base. One of the most common bases for a logarithm is base 10. When a logarithm has a 10 as the base, it's called a common logarithm. If a logarithm is written without a base - without a little subscript number written - you can assume that the base is 10.
Most calculators have a button for a common logarithm. Look for the "log" button on your calculator. If you type in log(100), it should say the answer is 2 because 10 to the second power equals 100.
Natural Logarithms - Base e
Another number that's commonly used as the base of a logarithm is the number e. e is an irrational number that comes up so often that it gets it's own symbol (similar to how the number 3.14159. . is called pi). The constant e is named after the mathematician Leonhard Euler. It is approximately equal to 2.71828. When a logarithm has a base of e, it's called a natural logarithm. This is abbreviated as ln (that's a lowercase L. l for log, n for natural).
Most calculators also have a button to calculate a natural logarithm. Look for the "ln" button on your calculator. Remember, it's a lowercase L for log and an n to indicate a natural log.
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