67 Solve Exponentials for Time: Logarithms
Earlier, we found that since Olympia, WA had a population of 245 thousand in 2008 and had been growing at 3% per year, the population could be modeled by the equation
Pn = (1+0.03)n (245,000), or equivalently, Pn = 245,000(1.03)n.
Using this equation, we were able to predict the population in the future.
Suppose we wanted to know when the population of Olympia would reach 400 thousand. Since we are looking for the year n when the population will be 400 thousand, we would need to solve the equation
400,000 = 245,000(1.03)n dividing both sides by 245,000 gives
1.6327 = 1.03n
One approach to this problem would be to create a table of values, or to use technology to draw a graph to estimate the solution.
From the graph, we can estimate that the solution will be around 16 to 17 years after 2008 (2024 to 2025). This is pretty good, but we’d really like to have an algebraic tool to answer this question. To do that, we need to introduce a new function that will undo exponentials, similar to how a square root undoes a square. For exponentials, the function we need is called a logarithm. It is the inverse of the exponential, meaning it undoes the exponential. While there is a whole family of logarithms with different bases, we will focus on the common log, which is based on the exponential 10x.
Common Logarithm
The common logarithm, written log(x), undoes the exponential 10x
This means that log(10x) = x, and likewise 10log(x) = x
This also means the statement 10a = b is equivalent to the statement log(b) = a
log(x) is read as “log of x”, and means “the logarithm of the value x”. It is important to note that this is not multiplication – the log doesn’t mean anything by itself, just like √ doesn’t mean anything by itself; it has to be applied to a number.
Example 9
Evaluate each of the following
a) log(100) b) log(1000) c) log(10000) d) log(1/100) e) log(1)
a) log(100) can be written as log(102). Since the log undoes the exponential, log(102) = 2
b) log(1000) = log(103) = 3
c) log(10000) = log(104) = 4
d) Recall that [latex]{{x}^{-n}}=\frac{1}{{{x}^{n}}}[/latex]. [latex]\log\left(\frac{1}{100}\right)=\log\left({{10}^{-2}}\right)=-2[/latex]
e) Recall that x0 = 1. log(1) = log(100) = 0
It is helpful to note that from the first three parts of the previous example that the number we’re taking the log of has to get 10 times bigger for the log to increase in value by 1.
Of course, most numbers cannot be written as a nice simple power of 10. For those numbers, we can evaluate the log using a scientific calculator with a log button.
Example 10
Evaluate log(300)
Using a calculator, log(300) is approximately 2.477121
With an equation, just like we can add a number to both sides, multiply both sides by a number, or square both sides, we can also take the logarithm of both sides of the equation and end up with an equivalent equation. This will allow us to solve some simple equations.
Example 11
a) Solve 10x = 1000 b) Solve 10x