# Solve Logarithmic Equation

• Jan 13th 2008, 11:27 AM
happydino1
Solve Logarithmic Equation
I need help solving this logarithmic equation:

$
2\log_{5}x - \log_{5}2 = \log_{5}(2x+6)
$

I tried "subtracting" the left side and then exponentiating everything with a 5 to cancel out the logs, but then I get to a prime polynomial. I'm stuck.
Help is much appreciated :)
• Jan 13th 2008, 12:06 PM
red_dog
Conditions: $x>0$ and $2x+6>0$
$\Rightarrow x\in(0,\infty)$
The equation can be written as
$\log_5x^2-\log_52=\log_5(2x+6)\Leftrightarrow \log_5\frac{x^2}{2}=\log_5(2x+6)\Leftrightarrow \frac{x^2}{2}=2x+6$
Solving the quadratic we get $x=-2$ and $x=6$
but only $x=6$ satisfy the conditions.