Expert: Josh - 5/10/2004

Question
I do not understand anything about logarithms.

1.) write as a single logarithm 8lnx-5lny
                 3x-15
2.) solve for x: e      -21=57
(that "e" is rasied to 3x-15 power.)

Answer
Just as division is the inverse of multiplication, logarithm is the inverse of an exponential function.

Jasmine, you need to remember these rules.

Rule1: log(x^a)=a*log(x)
Rule2: log(c)-log(d)=log(c/d)
or     log(c)+log(d)=log(c*d)

where "log" here represents the natural logarithm, i.e.,log to the base "e", where "e" is approximately 2.71.
log should be written as log_e, but people get lazy.
log_e can also be written as ln.

So, in Q1, 8*ln(x)-5*ln(y)=ln(x^8)-ln(y^(-5))
=ln[(x^8)/y^(-5)]

In Q2, as mentioned earlier, log simply "undoes" an exponential. For example, log_e(e^a)=a,....
another example, 2^4=16, so log_2(16)=log_2(2^4)=4.
Get the idea?

So, e^(3x-15)-21=57
Step1: Add 21 to both sides to isolate the exponential term from the constant.
e^(3x-15)=78
Step2: Take the anti-exponential (i.e., log_e) function
(3x-15)=log_e(78)
Step3: Add 15 to both sides
3x=log_e(78)+15
Step4: Divide throughout by 3
x=[log_e(78)+15]/3

Cheers.