- Hi everyone, in this
video we're gonna talk
about the log properties, all right.
There are three properties
here of logarithms.
Log of A times B equals log A plus log B.
Log of A divide B equals
log of A minus log of B.
And C times log A equals
log of A to the C power.
And with these properties,
the bases would remain consistent.
What I mean is, if that's a base of 10,
then this would have to be a base of 10
and this would have to be a base of 10.
Okay?
So I'm gonna start out with an example.
Let's say we have the expression
log base five
of 25 over the square root of x plus one.
And we desire
to expand this expression, right,
into more logarithms.
How can we do that?
Well notice property II, right,
this quotient here,
we can use property II.
So I can write this as log base five of 25
minus log of the square
root of x plus one.
And we expanded it,
and we can touch this up a little bit.
First of all, log base
five of 25, all right,
this is a value, all right.
Five to some power in math is 25.
Five to the second power is 25.
So this is a two.
And over here, the square
root of x plus one,
I can rewrite that as x
plus one to the 1/2 power.
Okay, yeah, square root of x plus one
can be written in exponential notation
as x plus one to the 1/2.
And according to property III, everyone,
I can just take this exponent
and scoot it to the front.
Yeah, take this out and
just put it right there.
So I can write this entire
expression right here
as two minus 1/2 log
of x plus one.
Okay?
So, let's do another example.
Let's do another example
where we have to expand
a logarithmic expression.
How about, let's see,
log of x to the third power
times the square root of y
all over z to the fourth power.
Let's expand this log expression.
All right we notice the quotient,
so we can use property II.
Rewrite this as log of
x cubed square root of y
minus log of z to the fourth power.
But are we done?
What I mean is can we expand this further?
Surely, right here.
Notice it's x cubed times
the square root of one.
We can use property I.
We can rewrite this as log of x cubed
plus the log of the square root of one.
And I'll rewrite this,
log of z to the fourth.
And we've expanded this,
and what else can we do?
Well according to this third property,
I can move that three to the front,
I can move this square root.
Square root of y is y to the 1/2 correct?
And I can move that four to the front.
So I can rewrite this as three log x
plus, remember that's y to the 1/2 power,
1/2 log of y
minus four log of z.
Okay?
Now let's go in reverse.
What if I gave you something
expanded already and I
want you to condense it
into one log?
All right.
How about, I'll use natural log,
the natural log of let's see
five the natural log of a
plus 1/2 natural log of b
minus four natural log of c.
So we've got five ln of a
plus 1/2 natural log of b
minus four natural log of c.
Can we take this
expression and condense it
into one log?
Well the first step is everyone,
in order to use these two properties,
there can't be anything in
front of that l of the log.
So to do this, if you're condensing this,
the first thing is, you need
to move this by property III.
Scoot that right there, take
this, scoot this right there.
So let's do that.
I'm gonna rewrite this.
This is natural log of a to the fifth
plus natural log of b to the 1/2 power
minus natural log of
c to the fourth power.
So we have to do that so if we wanna use
these two properties to condense it,
there's nothing in front of those logs,
there's no coefficient there.
All right, what can we do with these two?
Ah use property I.
That's the natural log of a to the fifth
b to the 1/2,
got this minus natural
log of c to the fourth.
Can we make this one log?
Ta dow, check that out, surely.
I'll just write natural log,
put a big parentheses
and put this over that,
a to the fifth power b to the 1/2 power
all over c to the fourth power.
We have now condensed
it into one logarithm.
And surely b to the 1/2 power,
can't we rewrite that
as the square root of b?
So I'm gonna write this again.
I can also write this as
natural log of a to the fifth
square root of b
all over c to the fourth.
Okay?
And last thing before we go,
we've been doing all this
stuff and it may seem silly,
but these properties are really important,
especially if you take calculus.
Because you'll be working a problem,
take you a few minutes,
and perhaps you'll get this answer.
Let's say you got the
answer natural log of 1/2,
and you look in the back of
the book to check the answer
and you see that the answer's
the natural log of two, okay?
A student who's not familiar
with these properties
may be stumped and think they
did the problem incorrectly,
but in fact these are equivalent.
These are the same values.
And we can tell by using these properties.
Check this out, see this right here.
Isn't this the natural log of one minus
the natural log of two?
And the natural log of one is zero.
So we have zero minus ln of two.
This is negative ln of two.
These are equivalent.
These are the same.
So knowing these properties, all right,
very very important.
That's it.
