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Everything you have learned in this chapter has built up to this moment right here. In previous sections, we learned the properties of exponents and the inverses of exponential functions. For the first time ever, you will be solving for the variable in an exponential equation which is usually sitting up there in the exponent. Now is your time to shine. Get your pencils and calculators ready!
Since we now know about exponential equations and their inverses, the natural log, we can easily solve for any variable in an exponential function. It's all about plugging in values for what we do know and solving for what we don't know. If the variable we are solving for happens to be in an exponent, we can apply the natural log to each side of the equation to help us solve.
Here is the general form of solving exponential equations:
y = ex
Now to get x out of the exponential function, we will apply the natural log to each side.
y = ex
ln(y) = ln(ex)
ln y = x
x = ln y
Let's go back to our bacteria problem.
Just for fun, let's revisit the water fountain bacteria problem we talked about in the Section "The ex Function." A bacteria is growing on a water fountain continuously and the number of bacteria is defined by the equation f(t) = 200e0.415t where t is the time in hours, and f(t) is the number of bacteria. How many hours will it take for the bacteria to triple?
Since we started with 200 bacteria, it would triple to 600 bacteria. We will plug those numbers into our equation to see what is left to solve for. (Glade commercial playing in the background: "Plug it in, Plug it in.")
f(t) = 200e0.415t
600 = 200e0.415t
At this point we need to divide each side by 200. Our goal is to get t by itself. It's not very sociable.
Now this is where our previous knowledge of inverses come into play. Take the natural log of both sides to get rid of the e function which seems to be holding our variable t hostage.
The world population grows continuously at about 1.915% a year. Due to the normal death rates, the overall population growth stays around 0.5%. Barring any population threat, like a zombie virus or some other pandemic, find how long it would take for the world population to double. Our equation will be P = e0.005t.
Since, we don't know exactly how many people are in the world, we can always make up some numbers, because in the end it's the final population divided by the initial population anyways. Let's assume there are 7 billion people in the world.
We may need some zombies if we have that many people living on earth. Experts say that they feel like population will stop growing and start declining, so we may be safe from over population.
What if we had a natural log function? We apply the same rules.
Solve ln(x2) = 121
This time we can raise each side of our equation as an exponent with base e since e is the inverse of ln.
Now that you've solved exponential equations, logarithmic equations will be a breeze. We will exercise the inverses of logarithms to solve for these, or possibly use a natural log. We will give you a few different ways to solve logarithmic equations. Once you've gone through this last section, you can put on your graduation cap. You will have graduated from this chapter of precalculus. Now, go log them trees!
Just like we can take inverses of exponential functions to solve equations, we can do the same process with logarithmic functions. If your function looks like this:
y = logbx
You can take the inverse of this equation to remove the variable x from the log. This is swiftly accomplished by taking the base, b, and raising each side of the equation as b's exponent.
y = logbx
by = blogbx
by = x
x = by
That would be a good pick up line: "Darlin', I'd like to break you out of that log cabin by raising you to my powers."
Solve for x:
6 = log2x
We will be trying to get the x by itself by raising each side of the equation to the exponent with our base being 2.
6 = log2x
26 = 2log2x
64 = x
x = 64
Solve for x:
log5(x + 1) = 6
We need to raise each side of this equation using the base of 5.
5log5(x + 1) = 56
x + 1 = 15625
x = 15624
Yay! We did it! (Well, we did, anyway. We don't know about you.)
Now, how long is it going to take you to pay off that car? Forever. Calculate how many years it will take to pay off your new car, which was $18,000 on the lot, if your monthly payments are $200 and the APR is 3.8%. This equation was introduced in our Exponential Money section.
Here is what we know:
PV = 18000
R = 200
r = 3.8% = 0.038
n = 12
t = ?
Substituting those values in:
First, divide each side by 350, then simplify more:
Subtract 1 from each side:
Since the base of the equation is 1.0031667, if we take the log of both sides using that base we get this:
log1.00316667 0.837143 = log1.00316667 (1.00316667)-12t
log1.00316667 0.837143 = -12t
-12t = log1.00316667 0.837143
Remember the little trick that we learned to compute logs (but switching them to natural logs)?
After about 5 years, you can have this car paid off. So don't start dropping french fries between the seats, you are going to be trying to keep this car nice to avoid car roaches.
This time, we are going to combine methods, because the natural log is SO versatile. Check it out...
Solve for x:
Did you notice that the (x + 3) was brought to the front? That's a nice little trick using one of the properties of logarithms. If you have a log or natural log that has an exponent, bring it out in front and multiply it. We think using natural logs is far easier than dealing with logarithms in problems like this. If you can remember to use the natural log whenever possible, solving these equations will be a breeze.