Logarithms have important uses in solving problems with complicated exponential equations. Consider the following example:
The population of a small town is 1000 onJanuary 1, 2001. It grows at aconstant rate of 2% per year. In whatyear does the population of thetown first exceed 1500?
This question is like the exponential growthproblems we’ve just seen but with a twist. Here, we’re given the growthrate, the initial quantity,
and the ending quantity. We need tofind the number of percent changes (in this case, the number of years)that links all these values. Since logarithms are the power to whichyou must raise a given number to equal another number, they are theperfect tool for solving this sort of problem.
In this case, it will take roughly 20.5years for the town’s population to exceed 1500. So about halfwaythrough the year 2021, the population will first exceed 1500.
The general form for a problem like thisone, in which the exponent is unknown, is to isolate the exponentialterm, take the logarithm of both sides, and then use the power rule oflogarithms to bring the variable out of the exponent. You can thenisolate the variable on one side of the equation. The base of thelogarithms is insignificant. You could choose a base-10 logarithm or alogarithm of any other base, as long as it is consistently used.
Here’s a simple example to illustrate this process:
If 6
x = 51000, then find the value of
x.
This problem would be vastly more difficult if we didn’t have logarithms. How would you possibly calculate 51000 anyhow? And how do you solve for
xwhen it’s the exponent of a number? But by taking the logarithm of eachside of the equation, and utilizing the power rule of logarithms:
The confusion clears, and we see that we have a logarithm problem that can be methodically solved.
