Math Logarithms

Question

The equation 7.35 = 14.7e^{– 0.0429h} can be solved to determine the elevation, h, in thousands of feet, where the air pressure is expected to be 7.35 pounds per square inch.

Find this elevation, h, in thousands of feet and round your answer to the nearest hundredth.

Answer 1

7.35 = 14.7e^{– 0.0429h}

^{Remember: a logarithm IS an exponent.. A natural logarithm (ln) is the exponent required on the number e (Euler's number) that is needed to give some value for a result. Your calculator can do this for you. but you have to understand what's going on to use it.}

^{First, let's simplify this equation a little then use it to show how a logarithm works.}

^{First, let's divide both sides by 14.7}

7.35/14.7 = (14.7e^{– 0.0429h)})/14.7

0.5 = e^{– 0.0429h}

Now it's in a simpler form and we have to solve for h. Since the h is in the exponent of e, simple algebra won't help.

Enter the natural logarithm.

For example, what power would we have to raise e to to get 2 for an answer?

That would be the natural logarithm of 2. This is the definition of the natural logarithm!

e^ln(2) = 2

What?! The calculator tells me that ln(2) = 0.693147...

So e^0.693147 = 2

Okay. So how can we use this to solve our problem?

0.5 = e^{– 0.0429h}

Here we have e raised to some power equals 0.5

That "power" is the exponent on e so it has to be the natural logarithm of 0.5.

A logarithm IS an exponent. A natural logarithm is an exponent on Euler's number e.

In this case 0.5 = e^{– 0.0429h}

e^{– 0.0429h} = 0.5 (I just swapped sides in the equation because it's more like how I explained a logarithm)

And, by definition:

e^ln(0.5)=0.5

ln(0,5) is the same thing as (is equal to) – 0.0429h

ln(0.5) = – 0.0429h

Your calculator can do the ln(0.5) so we can do one more step to solve for h.

Divide both sides of the equation by (– 0.0429).

ln(0.5)/(– 0.0429) = – 0.0429h/(– 0.0429)

ln(0.5)/(– 0.0429) = h

Enter the left side into your calculator and you get:

h = 16.1573

Logarithms are one of the most confusing things in math because they aren't like anything you've seen before. You have to remember that you're working with an EXPONENT, so some different rules apply: the rules of exponents! You'll see that soon enough.

This problem is as good as any to illustrate how logarithms work.You can go back through this problem multiple times, reading carefully and trying to see what's happening at each step.You'll start to get used to it after a while.

Natural logarithms (ln(x)) are exponents on Euler's number e.. e ^ln(x) = x

Common logarithms (log(x)) are exponents on 10. 10 ^log(x) = x

Your calculator can work with both types.

(There can be other base values besides e and 10. Don't worry about that yet. You don't need more confusion right now.)