>> This is more solving
for exponential equations.
So, before we had our
exponential equations,
we would change your bases
so that we can set
the exponents equal.
But with these, it's
not possible.
You can't turn an
11 into a base of 5
without doing a lot of work.
So, we use another technique and
we go back to what we learned
in sum and difference
of logarithms.
If you have a log in front of
this situation, you can take
that exponent and put
it right in front.
So, when we have these,
we put a log on each side,
so we take the log of the
left and the log at the right
and that allows us to take this
exponent and put it in front.
Now, we do not have an
exponential equation anymore,
we have an equation
that we can solve.
So, we divide by 3 log of 5,
plug it all in your calculator
and it gives us an answer.
That's it.
So, if you see that you cannot
make the bases the same then you
put a log on each side.
Now, you want to choose
your logarithm wisely.
So, I don't want to use
log here, I want to use LN
because LN is the base for
E-- has a base of E. So,
this is really log base E
which means it will allow me
to bring this down just like any
normal logarithm however I'm now
left with LN of E and what
is the natural log of E?
So, if you remember,
natural log of E is 1
because they cancel
each other, right?
So, you can just basically
throw it away because it's 1.
And there's our equation.
So, now we can just
divide by 0.06 and plug
that into our calculator.
So, we'll take LN of 30
and divide it by 0.006.
So, I have to reenter it.
LN of 30 divided by 0.006.
And there it is,
566.87 get it rounded.
566.87. All right.
So now what's left
is to check it.
So, we're going to take
this number, plug it back
into our exponential,
it's already entered
and I used the whole thing
to give me a better answer,
hit equals, and that's
basically 30.
And that's it.
So, you choose a
logarithm for each side.
If you pick the one with the
same base, it'll cancel it.
All right.
What if it's in this form?
So, it's not already set
up as 1X exponential.
So, what you have to do is
solve and I like to circle it
and you treat this one
just like it's the X. So,
that means you have 2,
add 2 to both sides.
And so we get 8 times 6X
equals 8 and then divide by 8.
So that's 6X-- 6 to the
power of X equals 1.
Now, we can put a
log on both sides.
And you could use the log base
6 but I don't want to enter
that into my calculator
so I'm going to leave it
as a plain log base 10.
And then log of 1 is 0,
and then we can take our X
and bring it in front.
And now we have our
nice linear equation.
So, we divide by log of 6
and we don't have to enter
that in the calculator
because it's 0, X is 0.
And how do we know it's correct?
Plug it back in.
6 to the 0 is 1,
and 8 minus 2 is 6.
Checks out, all right?
Let me do one more
and then you try some.
So, what if you have an
exponent on both sides?
So, then we go log of both
sides and that allows us
to bring the exponent in front
but make sure you put
a parenthesis around it
because there are two terms
there and there's our equation.
Now, what I'd like to do
is take this and divide
if off to the other side.
This allows me to not have
to distribute anything
and it gives me one number here.
So, we have X log
of 4 over log of 3.
And you have an option.
You can use the logarithm
and solve it completely
and get an exact answer or we
can just go for the approximate
which is what I'm going to do.
So, we're going to take log of
4 and divide it by log of 3.
So, log of 4 divided
by log of 3.
You could also make it
into a change of base.
There's a lot of
options but we're going
to use a bunch of decimals.
So, 1.26186, that's
what we're going to use.
1.26186. 1.26186X and we
double check, 1.26186.
There we go.
So, you could also have made
X and then log of 4 base 3
but it's going to get
complicated if we don't.
So, now we have a
nice linear equation
and we can just solve it
by subtracting X
and subtracting 1X.
That's all we have is 1X there.
And this gives us 0.26186X
and then we can just divide.
[ Pause ]
And the nice thing
with calculators is
that it remembers the value
so we can enter it in this way
and get a very good answer
which is 3.19, we'll say 3.19.
So, X equals 3.19, 189,
819, there we go, OK.
819, but if you don't,
you can just enter it
in the way we have it.
So, it's 1 divided
by 0.26186, 0.26186
and it's still the
same thing, 3.819.
So, as long as you have enough
decimals, it will work out
and give you a good
approximation.
Let's circle it and then
we're going to check it.
So, we bring up the calculator
and we go 3 exponent parenthesis
1 plus what we just got,
hit enter, and then
we do the same thing.
So, it's about 199.
So, now it's 4 to the exponent
of what we just got, 199.144.
See, same answer so we're good.
This is the correct answer.
All right, there are two
problems for you to try.
So, remember you need to
isolate your exponent first
and then try the logarithm and
then this one is ready to go.
So, hit pause and
give them a shot.
And there are the answers.
So, we had to isolate
the exponential first
and then we can put a log on
each side bringing down the 3X.
And so, this is just the 1X
we just go straight ahead
and divide, plug it in your
calculator, you get this.
Make sure you put a
parenthesis on the denominator.
If it's not giving you
that answer, it's probably
because you did not parenthesis
the entire denominator.
And then over here,
it's already isolated
so we can log both sides.
Bring down the 2X plus
1, but I didn't want
to distribute the log of 7
so I divided it over here,
plug that in my calculator,
it give me a decimal,
solve my equation,
and there's my answer.
And that's it.
