Is this the correct way to solve a logarithmic equation?

[lionely]

Homework Statement

2log₂ x - log₂ (x-3) = 2





The attempt at a solution

So what I did was , expand the brackets

2log x - logx + log3 = 2

logx = 2-log3

logx = log 4 - log 3

log x = log ( 4-3)

x= 4/3?

Is this right?

 

[Dick]

Homework Statement

2log₂ x - log₂ (x-3) = 2





The attempt at a solution

So what I did was , expand the brackets

2log x - logx + log3 = 2

logx = 2-log3

logx = log 4 - log 3

log x = log ( 4-3)

x= 4/3?

Is this right?

No, it's wrong. log(a-b) is not equal to log(a)-log(b). Try taking 2 to the power of each side of your original equation.

 

[lionely]

Aww got that wrong on my test then ;((

Okay how about this?

log x^2 / (x-3) = log 4

removing logs

x^2/(x-3) = 16
16x-48 = x^2

x^2 -16x + 48=0
(x -4 )(x -12)

 

[Dick]

Aww got that wrong on my test then ;((

Okay how about this?

log x^2 / (x-3) = log 4

removing logs

x^2/(x-3) = 16
16x-48 = x^2

x^2 -16x + 48=0
(x -4 )(x -12)

=/ messing up operations if the worst feeling in the world =/

What you get after 'removing logs' would be fine if your original equation were 2log(x)-log(x-3)=4 since 4=log(16). Was it?

 

[Mark44]

Aww got that wrong on my test then ;((

Okay how about this?

log x^2 / (x-3) = log 4

removing logs

x^2/(x-3) = 16
16x-48 = x^2

x^2 -16x + 48=0
(x -4 )(x -12)

=/ messing up operations if the worst feeling in the world =/

This is wrong, too, right in your first step. I should add that what you wrote is probably not what you meant. This is how the left side would be interpreted:
$$ \frac{log x^2}{x - 3}$$

From your subsequent work, I think this is what you meant:
$$ log(\frac{x^2}{x - 3})$$

If so, when you write it in plain text (as opposed to LaTeX) add parentheses, like this
log(x²/(x - 3))

Where you headed in your first step uses the idea that if log A = log B, then A = B.
Applying that idea, you get x²/(x - 3) on the left side. What do you get on the right side? Hint: NOT 16.

 

[lionely]

log(x^2/(x-3)) = log 4

(x^2/(x-3)) = 4?

 

[haruspex]

log(x^2/(x-3)) = log 4

(x^2/(x-3)) = 4?

Yes.