Logarithms are closely related to exponents and roots. A logarithm is the power to which you must raise a given number, called the base, to equal another number. For example, log2 8 = 3 because 23 = 8. In this case, 2 is the base and 3 is the logarithm.
The Math IC likes to use logarithms in algebra problems, mostly in simple equation-solving problems (which we cover in the next chapter). For any of these types of questions, the key thing to remember is that a logarithm problem is really an exponent problem. Keeping this in mind should help reduce the mystery that seems to surround logarithms. In fact, once you get the hang of it, you’ll realize that solving logarithmic equations is actually quite simple and easy.
Having defined logarithms in a sentence, let’s show one symbolically. The next three equations are equivalent:
For example, log
4 16 = 2 because 4
2 = 16 and
= 4. You should now be able to see why the three topics of exponents, roots, and logarithms are often linked together. Each method provides a way to isolate one of the three variables in these types of equations. In the example above,
a is the base,
b is the exponent, and
x is the product. Finding the root, logarithm, and exponent isolates these values, respectively.
Logarithms and Calculators
Unless the logarithm is a very simple one, you won’t be able to mentally calculate it—so the calculator becomes an important tool. But there is one important thing you need to be aware of. On your calculator, the LOG button assumes a base of 10. This means that for the equation log4 16 = 2, if you punched in LOG 16, you would get log10 16.
Some calculators can calculate a logarithm with any base you want, but less advanced calculators might not. In general, as long as your calculator is scientific, it should be able to calculate logarithms with different bases.
Calculate a few logarithms for practice:
Operations on Logarithms
You will rarely see a test question involving basic logarithms such as log10 100, or log2 4. In particular, on the logarithm questions you’ll see in the Algebra chapter, you’ll need to be able to manipulate logarithms within equations. So, you should know how to perform the basic operations on logarithms:
- The Product Rule: when logarithms of the same base are multiplied, the base remains the same, and the exponents can be added.
- The Quotient Rule: when logarithms of the same base are divided, the exponents must be subtracted.
- The Power Rule: when a logarithm is raised to a power, the exponent can be brought in front and multiplied by the logarithm.
You might have noticed how similar these rules are to those for exponents and roots. This similarity results from the fact that logarithms are just another way to express an exponent.
