Log question Algebra 2

Question

Algebraically determine the point of intersection of the two logarithmic fuctions given below.

 

 

y=log(subscript 3)(x-6) and y=3-log(subscript 3)x

Answer 1

The curves intersect when log₃(x-6) = 3 - log₃x

 

                                     log₃(x-6) + log₃x = 3

 

                                     log₃[x(x-6)] = 3

 

                                            x(x-6) = 3³

 

                                       x²-6x-27 = 0

 

                                       (x-9)(x+3) = 0

 

                                        x = 9 or x = -3

 

-3 does not satisfy the original equation.  So, x = 9.  

 

The curves intersect at the point (9, log₃(9-6)) = (9, log₃3) = (9,1).

 

Answer 2

Set the equal to each other and solve using log rules and algebra:

 

Log₃(x-6) = 3-Log₃(x)

 

Log₃(x-6) + Log₃(x) = 3

 

Log₃(x(x-6)) = 3  convert to an exponential equation

 

3³ = x² - 6x

 

x² - 6x - 9 = 0  and solve using the quadratic formula