so if I was looking at a problem like
this what I would focus on would be this
power okay that's what I'm going to want
to simplify first let's go ahead and
rewrite this out
all right so there's a couple properties
and if you remember you know one of the
big reasons why we spend so much time
simplifying logarithmic Expressions so
therefore we can realize this and say
hey this is kind of too complicated this
is too much is going on I want to
simplify this if I'm going to go ahead
and solve an equation looks like this
I'm going to want to be able to simplify
this so what first thing I recognize is
I can apply the power I can take this
number in front right that's being
multiplied and I can rewrite that as a
power
of my argument
okay so now you can simplify that and
remember the power rule of exponents x
squared to the Q power is going to be an
x to the sixth
and now if you remember the quotient
properties for logarithms whenever you
have a logarithm minus another logarithm
you can rewrite that as the quotient of
your two arguments
and again I'm just working on
my step here so now I can simplify this
to Ln of X cubed
all right so that's kind of important
here so I'm going to go ahead and now
simplify this to see what we get so we
have e raised to the Ln of X cubed equal
to an Ln e to the 16th property okay so
actually before I even write that in
there I think hopefully you can
recognize that you know when you have Ln
of remember that has a base e of e
raised to 16 power that is another
logarithmic property here we can just
simplify that to 16.
okay so now I have a much simplified
exponential equation right and if you
were to look at this probably this
original problem you probably might get
like overwhelmed or a little confused so
that's why it's so important and if you
notice like I did my work over here
because I wanted just to focus on I
wanted to you know not trying to do
everything in my head and you can do it
but I'm just telling you that's where
like I make mistakes that's where I see
a lot of my own students make their
mistakes when they try to just do
something in their head rather than just
like writing on a side sheet of paper
right so just take your time don't feel
like you always have to rush you know
something like this there was a lot of
steps that I did obviously as I was
teaching I'm explaining it but I want to
make sure you're following through on
everything there okay so now we have e
Raisa Ln of X cubed equals 16. so I need
to solve for x right that's what we're
trying to do we're trying to solve for x
so to find the value of x right whenever
it's in the power you've got to take the
loggable sides now again what log do we
take do we take the natural log do we
take log base 10 log base 5 we want to
always take it as the same base and
remember natural log has a base of e so
now I have Ln of e raised to the Ln of X
cubed is equal to an Ln of 16. okay now
hopefully you recognize here this is
again another rule of exponents whenever
you have log base e of e right and
that's raised to power then that's just
going to equal to that power and then
have wait a minute uh X cubed I don't
remember X
oh my God are you serious you guys did
the mistake I did I was dividing X6 X is
six divided by x squared that's you
don't divide them right what's the rule
of exponents you subtract the powers
when you have an exponent racing on the
power you multiply when you have
exponents divided by another X you
subtract the power so I am an idiot
ah
that is uh things that happen though
ladies and gentlemen so that's going to
be an x to the fourth
um so therefore these are all x to the
force
and oh man
okay there you go so be careful
um so now I'm going to have an Ln x to
the fourth is equal to an Ln of 16. so
now what we can do is apply the one to
one property right whenever we have a
logarithm equals a logarithm again both
are isolated logarithms with the same
base they have to be isolated and they
have to have the same base remember that
because that's a really common mistake
students will make is so you apply the
one-on-one property when the logarithms
are not isolated or the bases are
different but now I can just say x to
the fourth is equal to a16.
so now if I'm going to take the fourth
root so undo the fourth power right just
like you undo squaring you take the
square root or undo the fourth power
we're going to take the fourth root now
how do you simplify the fourth root of
16 and just remember also this is plus
or minus right so since it's a even we
have to do that plus or minus so let's
go ahead and rewrite this as the fourth
root of x to the fourth those are
inverse operations so that's going to
undo it so it's going to be plus or
minus and then now what I'm going to do
is say the fourth root of how do you
take the fourth root of 16. well the
best thing I would recommend doing is
again using your exponents to be able to
rewrite 16 as 2 to the fourth power then
you say x is equal to plus or minus A2
right because the fourth root of 2 to
the fourth are going to undo each other
so now we have x equals plus or minus 2
but we have to be careful we always want
to check for extraneous solutions
extraneous solution between it's a
solution to your equation to your final
it's a solution to your answer but it's
not a solution to your original equation
what I want you to recognize here is
negative two you cannot plug in negative
2 into the Ln of X because you can only
take the logarithm of positive values so
therefore x equals negative 2 is what we
call an extraneous solution so therefore
our only answer in this case is going to
be X is equal to a positive 2. so
hopefully this video was helpful for you
it gave you some value and if it did I
know you're going to join in the next
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