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 IIC likes to use logarithms in algebra problems, mostly in simple equation solving problems (which we will cover in the next chapter). For these types of questions, the key is remembering 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 easy.

Having defined a logarithm in a sentence, let’s show it symbolically. The three equations below are equivalent:

For example, log₄ 16 = 2 because 4² = 16 and = 4. You should be able to see now why the three topics of exponents, roots, and logarithms are often linked. Each 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. Respectively, finding the root, logarithm, and exponent isolates these values.

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 critical thing you need to be aware of. On your calculator, the “LOG” button assumes a base of ten. 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, but less advanced calculators might not. When you need to calculate a logarithm with a base other than 10 and your calculator does not have the capability, keep in mind that: log_b a = log_c a ⁄ log_c b.

Calculate a few logarithms for practice:

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. You should therefore know how to perform the basic operations on logarithms:

when logarithms of the same base are multiplied, the base remains the same, and the exponents can be added:

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.

when logarithms of the same base are divided, the exponents must be subtracted:

Natural logarithms appear on the Math IIC but not on the Math IC test. A natural logarithm is one with a base of e. The value e is a naturally occurring number, infinitely long, that can be found in growth and decay models. The natural logarithm will most likely be used in problems of growth and decay.

A common numerical approximation of e is 2.718, which you could easily discover by punching e¹ into your calculator. The symbol for a natural logarithm is ln, instead of log. The following three equations are equivalent:

Working with natural logarithms is just like working with logarithms; the only difference is that the base for natural logarithms is always e. You might also be asked to identify the graphs of ln x and e^x. Take a good look at their general shapes: