Question 554033: If log x^5y^3=25 and log x/y=3 what is log x?
Found 2 solutions by Theo, rapaljer:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if log(x^5y^3) = 25, this means that:
10^25 = x^5y^3
if log(x/y) = 3, this means that:
10^3 = x/y
we can solve for x in this equation to get:
x = 10^3y
we can substitute in the equation of:
10^25 = x^5y^3 to get:
10^25 = (10^3y)^5*y^3
this becomes:
10^25 = 10^15y^5*y^3
we simplify this to get:
10^25 = 10^15*y^8
we divide both sides of this equation by 10^15 to get:
10^25/10^15 =y^8
we simplify to get:
10^10 = y^8
we take the 8th root of both sides of this equation to get:
10^(10/8) = y which simplifies to:
10^(1.25 = y which leads to:
y = 17.78279410
since x = 10^3y which is the same as 1000y, this means that:
x = 17782.79410
y = 17.7829410
with those values for x and y, we get:
first equation:
log(x^5*y^3) = log(17782.79410^5*17.7829410^3) = log(10^25) = 25
second equation:
log(x/y) = log(17782.794/17.7829410) = log(1000) = 3
all the numbers check out.
the values for x and y are good.
x = 17782.79410
y = 17.7829410
log(x) = log(17782.79410) = 4.25
log(y) = log(17.7829410) = 1.25
your answer to the question is:
log(x) = 4.25
Answer by rapaljer(4671)  (Show Source):
You can put this solution on YOUR website! By laws of logarithms:
5 log x + 3 log y = 25
log x - log y = 3
Hmmm! A system of equations with logarithms! I've never seen that before!
Multiply each side of the second equation by 3:
5 log x + 3 log y = 25
3 log x - 3 log y = 9
Add the equations together:
8 log x = 34
Divide by 8:
log x = 34/8 or 17/4
For a NON-TRADITIONAL explanation of LOGARITHMS, please see my own website, but you won't find anything like this one! The easiest way to find it is to use the easy-to-remember and easy-to-spell link www.mathinlivingcolor.com. At the very bottom of this page there is a link that will take you to the Homepage of my website. I have a complete ALGEBRA curriculum there with LOTS of practice tests, and even a few videos. Best of all, it's all FREE!!!
For this particular topic, when you find the Homepage, look for the link "Basic, Intermediate, and College Algebra: One Step at a Time", choose "College Algebra" and look in "Chapter 4." There is a complete explanation, illustrated with the TI84, together with lots of examples and exercises with ALL the answers given. You will especially like the "Math in Living Color" pages that go with this, where hundreds of the hardest exercises are solved for you IN COLOR! I even have TWO videos of me teaching LOGARITHMS from before I retired. Remember, it's all FREE!
If you or anyone needs to contact me, my Email address is rapaljer@seminolestate.edu.
Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus
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