Exponential Equations

1
Exponential Equations

• Solving exponential equations requires the base
  on the left and right side to be equal. Consider
  the following example

A common mistake is to divide both sides by 16.
DONT DO THIS!
163x-51282x-4
First we need the same base on each side.
Now simplify the exponents on each side to get a
single power of 2 there.
212x-20214x-28
This creates a new equation with the same
solution as the original.
We can solve this like any other equation and get
the solution to the original exponential.
x4
2
Exponential Equations

• Lets check the previous question

Sub in 4 for x and see if the left and right side
are equal.
163x-51282x-4
163(4)-51282(4)-4
Now evaluate each side.
1671284
Thus x 4 checks out.
268435456 268435456
Lets apply this to a scientific example. The
half-life of a substance is the time it takes for
that substance to decay to one half of its
original mass. The general half-life formula is
given by
Where A is the final amount of the substance, Ao
is the original amount of the substance, t is the
total time and h is the half-life (same units as
the time)
3
Exponential Equations

• Consider a substance that currently has 200g. If
  it has a half-life of 4.8 days, how long will it
  take to reduce to 25g?

As with any question you should start with what
is given.
Given Ao200g h 4.8d A 25g t ?
Next write down the formula.
Now sub in all of the known values.
We want to solve for t, so we need to isolate the
exponential on the right first by dividing by 200
What new equation will this create?
We need to now find a power of ½ that equals
0.125.
Now sub it in
4
Exponential Equations
Now that the bases are equal, we can write down a
new equation that has the same solution as the
old one.
We had
Solving for t, we get our answer.
Finish the question off with a statement.
Thus it would take 14.4 days for 200g of this
material to decay to 25g.
5
Example Two
Cobalt-60 which has a half-life of 5.3 years, is
used extensively in medical radiology. The
amount left at any given time is given by
a) What fraction of the initial amount will be
left after 15.9 years?
b) How long will it take until there is only
6.25 of the original amount left?