Logarithim and exponent tutorial

I see all the time kids asking about the same questions so I will create this as to reference when basic methodology is needed to be taught

To start off basic exponent rules:

...NOTE

Using this in conjunction with the second rule we can see that

Now I will move onto logarithmic properties

NOTE: is used in lieu of other logarithims for its shortness....these rules apply to all logarithims...and for knowledge

and

A very important one for solving equations as well as using a calculator is the change of base theorem which states

Most useful example is

another thing you should know is

Now I will go over the way to solve two basic logarithmic equations

Case 1...this is one with a pre-existing log

first combien the logs to get

we want to isolate the x so we introduce the exponential function( ) to eliminate the log

so we have

taking the squareroot of both sides and using rules 5 and 6 in the exponent section we see the answer is

Case 2....introducing logs to eliminate exponents...

First thing we want to do is introduce a log to elminate the exponent...the log we introduce is always of same base as the exponent in this case 8..so we have

at this point I think it is prudent to discuss extraneous solutions

is defined only for positive values for x...so you might get an algaebraic solution that works but it makes a log undefined..these solutiong are to be discarded

any questions just ask