Using logarithms to solve exhibit exponential
equations, part two - First let's review our
procedure.
We are going to isolate our exponential expression
and then look at the base.
If we have base 10, then we will take the
common log of both sides and if we have a
base other than 10, we're going to take the
natural log of both sides and then we will
simplify by using one or more of the properties
– log properties, of course – and then
we're going to solve for the variable.
So for example, let's look at 8 times 2 to
the 3X minus 4 plus 1 equals 25.
Now first we have to isolate the exponential
term, so we're going to start with subtracting
1 from both sides, and we get 8 times 2 to
3X minus 4 equals 24.
And then we will divide through by 8, and
we would get 2 to the 3X minus 4 equal 3.
And since those are not the same base – right
there – we're going to take the natural
log of 2 to the 3X minus 4 equals the natural
log of 3.
Now our next step is to use the power rule
for logarithms, but our exponent has more
than one term.
So when we bring it out in front it must be
in parentheses – so 3X minus 4 end parentheses
times natural log of 2 equals natural log
of 3.
Now I would advise against multiplying through
by natural log of 2.
I would go ahead and just divide by natural
log of 2 – and we get 3X minus 4 equals
the natural log of 3 divided by the natural
log of 2, and then add 4 – so I'm going
to have 3X equals natural log of 3 over natural
log of 2 plus 4 – and then multiply both
sides by the reciprocal of your coefficient
– in other words, 1/3, and don't forget
to put parentheses on the right here.
So what we have is 1/3 times 3X equals 1/3
times end parentheses natural log of three
divided by the natural log of 2 plus 4.
Now we're going to cancel out the 3s on the
left, given as our X, and we're going to distribute
the 1/3 through the parenthesis here.
So we're going to get natural log of 3 over
3 natural log of 2 plus 4/3 and then when
you put that into your calculator, round to
the fourth decimal place, you get 1.8617.
Now let's look at one with base 10.
Say that we had 5 times 10 to the X plus 2
minus 3 equals 0.
Now we're going to add 3 to both sides, so
5 times 10 to the X plus 2 equals 3 and divide
through by 5.
So 10 to the X plus 2 equals 3/5.
And then since this is base 10, we're going
to take the common log of both sides – common
log of 10 to the X plus 2 equals the common
log of 3/5.
Now base of the common log, of course, is
10.
So common log of 10 to the X plus 2 simplifies
by the inverse property of logs to X plus
2, and of course common log of 3/5 on the
right is a number.
And then so finally, we subtract 2 from both
sides and we're going to get X equals common
log of 3/5 minus 2, and you put that into
your calculator, and you get negative 2.2218.
Okay, now for our last example let's do one
that students usually have difficulty with
– and that is when you have an exponential
term on both sides, like 4 to the X plus 3
equals 3 to the 2X.
And notice the bases are different.
Now for this one you cannot isolate the exponential
term; it's just not possible.
There's two different ones.
So we're going to go straight to taking the
natural log of both sides.
So the natural log of 4 to the X plus 3 equals
the natural log of 3 to the 2X.
And then use the power rule for logarithms.
Now the X plus 3 exponent here is two terms,
so it must go in parenthesis again – X plus
3, now it's like a 4, and then over here on
the right that's just 2X natural log of 3.
Now this time because we are trying to isolate
for X – in other words, solve for X – we
are actually going to distribute the natural
log of 4 through the parenthesis.
So X natural log of 4 plus 3 natural log of
4 equals 2X natural log of 3, and I am going
to subtract X natural log 4 from both sides.
So I'm going to have 3 natural log of 4 equals
2X natural log of 3 – 3, I don't know why
I wrote X there; let's fix that real fast
– and minus X natural log of 4.
Okay, now over here on the right, I have X
as a common factor.
So we're going to factor that out.
So over here – 3 natural log of 4 on the
left stays the same.
Now I'm going to factor out X and I get 2
natural log of 3 minus natural log of 4.
Now there are three log properties – the
power rule, which we use a lot, but there's
also the product rule of logs and the quotient
rule of logs.
We could use the power rule and the quotient
rule to clean this up a little bit.
Over here on the left, let's use the power
rule and make this natural 2 log of 4 cubed
–equals X times.
Now let's use the power rule also in this
first term inside the parenthesis – so natural
log of 3 squared minus the natural log of
4, and then that would give me natural log
of 64 equals X times – and inside the parenthesis
is a minus sign, so I could actually use the
quotient rule here and write that as the natural
log of, well 3 squared is 9 over 4.
And this just makes solving for X much easier.
So now we divide by natural log of 9/4 and
we get X equals natural log of 64 over the
natural log of 9/4, and when you put that
into your calculator you get 5.1285.
The end.
