Solving Logarithmic Equations

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 = log_bx

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 = log_bx
b^y = b^log_bx
b^y = x
x = b^y

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."

Sample Problem

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 = 2^log₂x
64 = x
x = 64

Sample Problem

Solve for x:

log₅(x + 1) = 6

We need to raise each side of this equation using the base of 5.

5^{log₅(x + 1)} = 5⁶
x + 1 = 15625
x = 15624

Yay! We did it! (Well, we did, anyway. We don't know about you.)

Sample Problem

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:

log_1.00316667 0.837143 = log_1.00316667 (1.00316667)^-12t
log_1.00316667 0.837143 = -12t
-12t = log_1.00316667 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.

Sample Problem

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.