+0  
 
+5
79
2
avatar

What is log base9 (x^2-4x) = log base9 (3x-10)

Guest Feb 27, 2017
Sort: 

2+0 Answers

 #1
avatar
0

Solve for x:
(log(x^2 - 4 x))/(log(9)) = (log(3 x - 10))/(log(9))

Subtract (log(3 x - 10))/(log(9)) from both sides:
(log(x^2 - 4 x))/(log(9)) - (log(3 x - 10))/(log(9)) = 0

Bring (log(x^2 - 4 x))/(log(9)) - (log(3 x - 10))/(log(9)) together using the common denominator log(9):
-(log(3 x - 10) - log(x^2 - 4 x))/(log(9)) = 0

Multiply both sides by -log(9):
log(3 x - 10) - log(x^2 - 4 x) = 0

log(3 x - 10) - log(x^2 - 4 x) = log(3 x - 10) + log(1/(x^2 - 4 x)) = log((3 x - 10)/(x^2 - 4 x)):
log((3 x - 10)/(x^2 - 4 x)) = 0

Cancel logarithms by taking exp of both sides:
(3 x - 10)/(x^2 - 4 x) = 1

Multiply both sides by x^2 - 4 x:
3 x - 10 = x^2 - 4 x

Subtract x^2 - 4 x from both sides:
-x^2 + 7 x - 10 = 0

The left hand side factors into a product with three terms:
-(x - 5) (x - 2) = 0

Multiply both sides by -1:
(x - 5) (x - 2) = 0

Split into two equations:
x - 5 = 0 or x - 2 = 0

Add 5 to both sides:
x = 5 or x - 2 = 0

Add 2 to both sides:
Answer:      x = 5            or              x = 2(assuming a complex-valued log)

Guest Feb 27, 2017
 #2
avatar+76821 
0

Since the log bases are the same, we can forget the log part and solve this :

 

x^2 - 4x   = 3x - 10       subtract 3x from both sides....add 10 to both sides

 

x^2 - 4x - 3x + 10  = 0     

 

x^2 - 7x + 10  = 0     factor

 

(x - 5) (x - 2)  = 0

 

Set both factors to = 0   and x = 5   or x  = 2

 

However.....we must reject 2 because it makes both logs undefined.....so.....

 

x = 5  is the solution

 

 

cool cool cool

CPhill  Feb 27, 2017

9 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details