What are logarithms? Well, to start, the word itself is a bit awkward at first. When students are first presented with the concept of these "logs," it is often part of their initial exposure to how exponents, or powers, are used. A logarithm is simply an exponent presented as something other than a superscript.
Once students have seen a few examples of logarithmic expressions, what tends to trip them up is the use of a base other than 10 in the log expression, which is the default value.
For example, if you were asked to solve the expression y = log21,000, there is no easy intuitive way to approach the problem.
Confused? Read on, and any "power" log expressions with non-standard bases have over you will disappear.
Logarithmic Expressions Explained
Say you are asked to solve the expression y = log101000. First, you need to identify what is happening in the problem. When you get a value for y, it has to be an exponent.
To be precise, it is the exponent (or power) to which the base (given as a subscript and taken to be 10 when not explicitly given) must be raised to get the argument of the log, which is the only number you see in standard form at the start of these problems.
That is, the above expression is equivalent to 10y = 1,000. You may recognize on sight that y must be equal to 3, but if not you can rely on your calculator to get the correct answer.
Why Use Logarithms, Anyway?
Why is it useful to look at the relationship between one number and the log of a second number instead of just examining and graphing the relationship as it is?
The answer lies in the fact that when y varies with some positive power of x, it increases more quickly than x does; as this power becomes even slightly larger, the increasing gap between x and y with increasing values of x becomes extreme. Because of this, it is common in such situations to graph y versus logbx or a constant multiplier of logbx.
- An example of this is the Richter scale in geological science, used to quantify the strength of earthquakes. Each whole-number step up the scale corresponds to a tenfold increase in magnitude as well as a 31-fold increase in energy released. Because of this, a quake with a magnitude of 7.7 releases 31 times the energy of a 6.7-magnitude quake and (31× 31 = 961) times the energy of a 5.7-magnitude quake.
Examples of Logarithmic Problems
Given y = log10100,000, what is y?
y is the exponent to which 10 must be raised to get the value 100,000. This is 5, as you may be able to do in your head if you know that 105 = 100,000.
Given y = log1050,000, what is y?
y is the exponent to which 10 must be raised to get the value 50,000. Clearly, this is a noninteger value since 104 = 10,000 and 105 = 100,000. You calculator can provide the answer: 4.698. (This is a good reminder that exponents do not have to be whole numbers.)
Log2x in Action
When you explore log problems with bases other than 10, none of the aforementioned principles change. The math can look a little wonkier, so take care not to confuse small bases like 2 with whatever the log is, as these numbers are often in the low single digits, too.
Example: What is log24,000?
The answer completes the sentence "4,000 is the result of 2 being raised to the power of..." The value of this expression is 11.965.
- You can use an online tool like the one in the Resources instead of your calculator to solve log2 problems.