Solve for x

Question

1. Log₆(x-1)+ log₆ (x-2) = 2log₆ √6

Answer 1

log(6)(x-1)+log(6)(x-2)=2log(6)√6

log(6)(x-1)+log(6)(x-2)=2log(6)(6)^(1/2)

log(6)[(x-1)(x-2)]=(1/2)(2)log(6)6

log(6)[(x²-x-2x+2]=1 (1/2)2=1 and log(6)6=1 also

convert to exponential form

6¹=x²-3x+2

x²-3x+2-6=0

x²-3x-4=0

(x-4)(x+1)=0

x-4=0

x=4

Answer 2

log₆(x-1) + log₆(x-2) = 2 log₆(√6)

 

Use the following log identities:

• log a + log b = log(a*b)
• log(ab) = b*log(a)

 

log₆((x-1)(x-2)) = log₆((√6)²)

log₆((x-1)(x-2)) = log₆(6)

 

Set the arguments of the logs equal:

 

(x-1)(x-2) = 6

x² - 3x + 2 = 6

x² - 3x - 4 = 0

 

Factors to:

 

(x-4)(x+1) = 0

x = 4 and -1

 

Check:

log₆(4-1) + log₆(4-2) = log₆((√6)²)

log₆(3) + log₆(2) = log₆(6)

log₆(3*2) = log₆(6)

log₆(6) = log₆(6)   Check for x=4

 

log₆(-1-1) + log₆(-1-2) = log₆(√6 2)
log₆(-2) + log₆(-3) = log₆(6)
log₆(-2*-3) = log₆(6)
log₆(6) = log₆(6)     Check for x = -1