5.4 Properties Of Logarithms

There are three very important laws of logarithms

Laws of Logorithms

Log a (uw) =log a (u) + log a (w)

Log a (u/w)= log a (u) - log a (w)

Log a (u^c) = c log a u. For every real number c

But remember, logarithms can be sneaky, and aren't always what you think. For example

Log a (u+w) Does NOT = log a u + log a w

Also

Log a (u-w) does NOT = log a u- log a w

How to use the laws of Logorithms

There are two ways to express Logorithms. One is as one Logorithm and the other is in terms of variables.

In order to change from one Logorithm into variables you would take the exponent of the variable and make it the coefficient in front of log.

For example you would change

log a x^3 into 3 log a x.

Remember, the whole point is to use the three laws of Logorithms.

The first step in changing more complex problems is to make sure that you have taken out all the multiplication and division, and replaced it with addition and subtraction.

The next step is to change the exponents to the coefficient, as we did earlier.

Now, for expression as one Logorithm.

The first step is just the inverse of last time. Here we change

3 log a x into log a x^3.

Next we will do the opposite again. Changing. From addition and subtraction into multiplication and division.

Next we solve logarithmic equations.

Log a (2x+3)= log a 11+ log a 3

Now remember- Log a (uw) =log a (u) + log a (w)

I told you Logorithms are sneaky

Because of law one the next step is

2x+3=33

So

X=15

How to solve ane equation like

Log 2 (x+2)= 4

Since you can't just get rid of the log 2's you have to do something with them.

It is important to remember that log 2 of 4 = 2^4. All you have to do is change it to exponential form.

So

X+2= 2^4

X+2=16

X=14

Graphs of logarithmic equations

When graphing log a x^2 it is the same except is reflected about the y axis, in addition to being where it is now.

Matthew "Seaglass" Silbergleit