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# find the exact solution to the equation below

log (x3)+log (x4) / log(100x) = 5
x=

### 1 Answer by Expert Tutors

Shannon H. | Mathematics and Science TutorMathematics and Science Tutor
4.9 4.9 (347 lesson ratings) (347)
1
Hi Dalia, so here it goes
Steps
log(x3) + log(x4)/log(100x) = 5                      Multiply log(100x) on both sides
log(x3) + log(x4) = 5log(100x)                       logM + logN = logMN, M = x3, N = x4
log[(x3)(x4) = 5log(100x)                              Add exponents on left hand side
log(x3+4)=5log(100x)                                    rlogM = logMr
log(x7) = log(100x)5                                     Bring 100x to the 5th power using (as)t = as x t
log(x7) = log10,000x5                                   If logM = logN, then M=N, M=x7, N=10,000x5
x7 = (1x1010)x5                                           Set equation to 0 by subtracting (1x1010)x5 on both sides
x7 - (1x1010)x5 = 0                                           Take out Greatest Common Factor(GCF) of x5
x5(x2 - (1x1010)) = 0                                         Set each part equal to 0 and solve
x5 = 0                      ;      x2 - (1x1010) = 0
(x5)1/5 = (0)1/5        ;            +1x10^10      1x10^10
x = 0                       ;       (x2)1/2=(1x1010)1/2
x = ±100,000
So, x =0,100