In this video we're going to look at some application problems for logarithms
and we're gonna start by looking at what's called common logarithms
common logarithms are logs with base 10
if the base is not written in a log, it's assumed to be 10
so, you'll see on your calculator that there's a log button
some of the like newer TI calculators, with the log button you can input the base
but on most calculators, the log button does not allow you to input the base
so if the log button does not allow you to input the base
then that means it's doing common logs, or logs with base 10 only
so we're going to look at an example here
and a lot of times in word problems when you have like a formula that's modeling something
they're gonna use logs with base ten, or common logs
so you won't see a base like in this example
so this is example eight
it says the percentage of adult height attained by a boy who is x years old can be modeled by
f of x equals 29 plus 48.8 times log of x plus 1
where x represents the boys age and f of x represents the percentage of his adult height
approximately what percentage of his adult height has a boy attained at age eight?
so this is really just a plugging-in problem
so we want to know what percentage of adult height the boy has at age eight
x is age, so we're gonna plug in 8 for x
so we're gonna calculate f of eight
which is gonna be 29 plus 48.8 times log of 8 plus 1
now one thing with doing logs on the calculator, is that you have to be careful about the parentheses
so I'm gonna show how to type this in on the Desmos calculator here
on this calculator, there's no log button on the main screen
when I click over here on the function button, I see the log button pop-up
so I'm gonna type in 29 plus 48.8
and then times, or I don't need to hit the times, it'll automatically do times
log, and then you see it opens up a pair of parentheses there
now, I can type eight plus one into the parentheses, or I can do eight plus one in my head
and I know that it's just 9, then I'm gonna close the parentheses after that
so type that in, and I get 75.567, there's a lot of numbers after the decimal point there
so, approximately 75.567 percent
this problem it actually doesn't tell me how to round
when you see this on the homework, it's gonna tell you to round to the nearest tenth
so that's what I'm gonna do here
in the book they round to the nearest whole number, just remember to look for that
on MyMathLab, that instruction for how to round
so if I round to the nearest tenth, it would round to 75.6 percent
so next you should look at checkpoint problem number eight
it's using the same function, but just finding the percentage at age ten
so do this in your notes, check your answer, and then move on with the rest of the video
next we're going to talk about natural logarithms
natural logarithms are logs with base e
so if we have log base e of some number, I can also write that like this
ln, and I usually use the cursive l, but you can also write a more of a print l
so ln of x is log base e of x
the ln stands for natural log
I think in some other language, because that's why the l comes before the n, and it's not nl
so going back to some of these that we wrote all the way back here
if I have log base e of 9 equals y
and you saw this if you looked at the notes, the other way that I can write this is ln of 9 equals y
so we're going to look at example 11
and this is an example of a function that's written with the natural log
we have the function f of x equals 13.4 times ln of x minus 11.6
models the temperature increase f of x in degrees Fahrenheit after x minutes
use the function to find the temperature increase to the nearest degree after 50 minutes
so x is minutes, so all I'm gonna do here is plug 50 in for x
so I'm gonna calculate f of 50
which is going to be 13.4
now, you have to be careful here because on a calculator, after the ln
it's going to automatically open up a parenthesis
you're gonna type in the 50, because 50 is x, and then you need to close the parentheses after the 50
if you don't do that, it's gonna do 50 minus 11.6 and then take the natural log of that number
which is not what you want to do
so closing parenthesis here is really important in getting the right answer
so I'm going to type this into my calculator
so we have 13.4
the ln button, on this calculator, there actually is an ln button on the main screen
and it also appears on the function menu
so ln 50, and then close the parentheses, and then minus 11.6
so the answer that I get here is 40.8211
and looking in the problem, it says to find the temperature increase to the nearest degree
which means round to the nearest whole number, so that would round up
40.8 would round up to 41 degrees
so next you should do checkpoint problem number 11
do this in your notes, check your answer, and then move on to the homework for this section
