(a) 6^2x = 250 (b) log3 (x-3) + log3 (x-7) = 4

Question

Round answers to four decimal places.

Answer 1

 

2x = log6 (250)

 

Changing the base:  log6 (250 ) = log10(250)/log10(6) 

 

Dividing that by 2:

so X = 1.540793 approximately

 

 

(B)

 

   log(AB) = logA + logB

 

  So log3( (x-3)(x-7) = 4

 

    (x-3)(x-7) = 3^4

 

   (x-3)(x-7) = 81

 

   x^2 - 10x + 21 = 81

 

  x^2 - 10x - 60 = 0

 

  x = [10 +or- sqrt( 100 - 4*60) ] / 2

 

  x = [ 10 +or- sqrt(-140)]/2

 

    = [ 10 +or- 2i*sqrt(35)]/2

 

   = 5 +or- i*Sqrt(35)

Answer 2

6^2x=250

log6^(2x)=log250

2xlog6=log250

2x=log250/log6

use your calculator to find these log values

2x=2.39794/0.77815

2x=3.08159

x=3.08159/2

x=1.5407955

 

log(3)(x-3)+log(3)(x-7)=4

log(3)([x-3][x-7])=4

3⁴=(x-3)(x-7)

81=x²-10x+21

x²-10x+21-81=0

x²-10x-60=0

use the quadratic formula

(10±√[100+240]/2

(10±√340)/2

(10±2√85)/2

5±√85

x=5+√85