Real World Math Part 3 – What are Logarithms good for?
In the previous article about depreciation, we introduced the exponential depreciation model. It is an excellent tool for use in a variety of fields. However, when applied to depreciation of monetary value over time, such as our automobile example, we discovered the difficulty of working with variables when they show up as an exponent.
Guess and check put us in the ball park for determining when our example vehicle – the Toyota Corolla MSRP at $21,000 financed over 5 years, has a value less than the cost of financing the car.
Instead of using the ball park method, there is a more elegant way to calculate a precise interval of time before our Toyota’s value is less than the finance charge. The following should give you some appreciation for why I like having logarithms available in my real world math tool kit.
First of all, logarithms are used to convert equivalent values between Log expressions and exponential expressions. For example:
Say we have an exponential expression: 1,000=10x When this expression is translated into a logarithmic expression, it it will look like this: log(1000)=3
When using the “Log” symbol on most calculators, it implies the log base is 10:
log10 1000 = 3
Our exponential decay model in the depreciating Toyota example follows:
$21,000 x 0.8(yrs) < $6,189 Simplified to – 0.8(yrs) < 0.2947
This inequality is still in exponential form with the variable in the exponent, so let’s convert it to logarithmic form:
log0.8(0.29) = yrs
But wait, my calculator doesn’t allow me to change the base of the logarithm from 10 to 0.8!
No worries, there is a formula for this very specific situation and it is called (would you believe?), the “change log formula.” As applied to our problem we get the following, which we can type into our calculator to get the exact amount of years:
log(0.29)/log(0.8) = 5.54 yrs
As you can see, the mathematicians of old have done a lot of hard work figuring things out for us. There are more situations where this relationship between exponential expressions and logarithmic expressions can be useful. Perhaps the most important point of this real world math exercise is the understanding that numbers, mathematical operations, ratios and formulas can be related to one another in a variety of ways. Filling your math tool kit with knowledge of these relationships can save lots of time spent wondering how to solve real world math problems.
