Exponential Equations Solved by
Logarithms (6.8)
Finding the value of the exponent when bx = a.
A little POD
Before, we solved equations like this: 102 = x. We found the argument.
Or like this: x2 = 49. We found the base.
Today we find the exponent.
Estimate the value of x:
10x = 3
x needs to be between which two numbers? How could you guess an
answer? Use your calculator to guess and check.
What is the base? The argument?
Exponential equations-- guess and check
1.
2.
3.
8x = 64
3x = 27
2x = ½
Getting the hang of it?
4.
64x = 8
Try this by thinking a minute first.
Exponential equations-- common bases
64x = 8
How can we solve this using a common base?
What happens if we change the problem?
64x = 4
64x = 16
8x = ¼
Keep looking for that common base.
Exponential questions-- common bases
Find the common base for these.
144x = 1728
512x = 4
Notice how we’ve been solving for the exponent in each
of these.
Exponential equations-- common bases
If 3x = 32, then what do we know about x?
If 5x-3 = 55, then how can we solve for x?
If 5x-3 = 25, what do we do first?
We can work each of these problems by using guess and
check, or thinking it out a little. We can also do it by
setting each side to a common base.
Let me show you one of them.
An easier way to solve for exponents
What happens when we can’t find a common base easily? Use logs!
If 10x = 3,
then x = log10 3
In general,
if x = 10y, then y = log10x = log x.
So, if 5x = 12,
then x = log5 12
In more general,
if x = by, then y = logbx.
See the pattern? Your calculator does logs base 10 very well.
Solve our POD and others using logs
now
If 10x = 3, then x = log103
1.
2.
3.
10x = 3
10x = 457
10x = 392
Solve our POD and others using logs
now
Sometimes you have to use a little algebra with logs.
1.
5(10x) = 216 (Divide by the number in front of the base first!)
2.
10.6x = 200
3.
10-x = 39 (Try this two ways.)
4.
15(10.3x) = 157
Check your answers!
Handy tools
Keep these log formulas handy for use:
log 10 = 1
log 100 = 2
log 1000 = 3
log 10x = x
so that log 103 = 3 (Test it on your
calculators.)
Try a little Change of Base
How might you solve this equation when the base is 6, not
10?
6x = 152
We’re still solving for an exponent, so let’s look at logs.
x = log6152
But your calculator doesn’t do base 6 logs. We have to set
this up differently.
Try a little Change of Base
How might you solve this equation when the base is 6, not
10?
6x = 152
x = log6152
Now, we can divide and solve.
x = (log 152)/(log 6)
This is a short-cut, called the change of base formula.
See how it works in the problem?
Try a little Change of Base
Try the same pattern with these problems:
5x = 13
7x = 49 (Check this one to see if it
makes sense.)