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 inalgebra problems, mostly in simple equation-solving problems (which wecover in the next chapter). For any of these types of questions, thekey thing to remember is that a logarithm problem is really an exponentproblem. Keeping this in mind should help reduce the mystery that seemsto surround logarithms. In fact, once you get the hang of it, you’llrealize that solving logarithmic equations is actually quite simple andeasy.
Having defined logarithms in a sentence, let’s show one symbolically. The next three equations are equivalent:
For example, log4 16 = 2 because 42 = 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 away 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 animportant tool. But there is one important thing you need to be awareof. On your calculator, the LOG button assumes a base of 10. This meansthat for the equation log4 16 = 2, if you punched in LOG 16, you would get log10 16.
Some calculators can calculate a logarithmwith any base you want, but less advanced calculators might not. Ingeneral, as long as your calculator is scientific, it should be able tocalculate 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 log24. In particular, on the logarithm questions you’ll see in the Algebrachapter, you’ll need to be able to manipulate logarithms withinequations. So, you should know how to perform the basic operations onlogarithms:
- 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 theserules are to those for exponents and roots. This similarity resultsfrom the fact that logarithms are just another way to express anexponent.
