SOLUTION: Given that P>1 and {{{1/log(2,P)}}} + {{{1/log(3,P)}}} + {{{1/log(5,P)}}} + {{{1/log(7,P)}}} + {{{1/log(11,P)}}} = {{{1/log(x,P)}}}, find the integral value of x

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Given that P>1 and {{{1/log(2,P)}}} + {{{1/log(3,P)}}} + {{{1/log(5,P)}}} + {{{1/log(7,P)}}} + {{{1/log(11,P)}}} = {{{1/log(x,P)}}}, find the integral value of x      Log On


   

Question 1204649: Given that P>1 and 1%2Flog%282%2CP%29 + 1%2Flog%283%2CP%29 + 1%2Flog%285%2CP%29 + 1%2Flog%287%2CP%29 + 1%2Flog%2811%2CP%29 = 1%2Flog%28x%2CP%29, find the integral value of x
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

Change of base formula
log%28b%2C%28x%29%29+=+log%28%28x%29%29%2Flog%28%28b%29%29
the reciprocal of that is
1%2Flog%28b%2C%28x%29%29+=+log%28%28b%29%29%2Flog%28%28x%29%29
This idea is used in the 2nd step shown below.



Use the idea mentioned earlier



log%28%282%2A3%2A5%2A7%2A11%29%29%2Flog%28%28P%29%29+=++log%28%28x%29%29%2Flog%28%28P%29%29 Use the log rule log(A)+log(B) = log(A*B)

log%28%282310%29%29%2Flog%28%28P%29%29+=++log%28%28x%29%29%2Flog%28%28P%29%29

log%28%282310%29%29+=++log%28%28x%29%29

x+=+2310

Answer by ikleyn(52767) About Me  (Show Source):
You can put this solution on YOUR website!
.
Given that P>1 and 1%2Flog%282%2CP%29 + 1%2Flog%283%2CP%29 + 1%2Flog%285%2CP%29 + 1%2Flog%287%2CP%29 + 1%2Flog%2811%2CP%29 = 1%2Flog%28x%2CP%29, find the integral value of x
~~~~~~~~~~~~~~~~~~~~~ 

Use an identity

    1%2Flog%28a%2Cb%29 = log%28b%2Ca%29,

which is valid for any positive real "a" and "b".


Then your equation will take the form

    log%28P%2C2%29 + log%28P%2C3%29 + log%28P%2C5%29 + log%28P%2C7%29 + log%28P%2C11%29 = log%28P%2Cx%29.


It is the same as

    log%28P%2C%282%2A3%2A5%2A7%2A11%29%29 = log%28P%2Cx%29,


which you can transform further

    log%28P%2C%282310%29%29 = log%28P%2C%28x%29%29

    x = 2310.


ANSWER.  x = 2310.

Solved.