# Tough logarithmic equation

• May 11th 2011, 09:42 AM
Arka
Tough logarithmic equation
Here's the equation which I have to solve for x,

$\log_5 (5^{\frac{1}{x}} + 125 ) = \log_5 (6) + 1 + \frac{1}{2x}$

I have tried making the RHS into a single logarithm to the base 5 and then taking anti-log......but that didnt help much.

• May 11th 2011, 10:04 AM
red_dog
$\log_5\left(5^{\frac{1}{x}}+125\right)=\log_56+ \log_5 \left(5^{1+\frac{1}{2x}}\right)$

$\log_5\left(5^{\frac{1}{x}}+125\right)=\log_5\left (6\cdot 5\cdot 5^{\frac{1}{2x}}\right)$

Then $5^{\frac{1}{x}}+125=30\cdot 5^{\frac{1}{2x}}$

Substitute $5^{\frac{1}{2x}}$ with $y$.

Can you continue?
• May 11th 2011, 10:08 AM
Ackbeet
Write the RHS all as log base five quantities. Then the RHS will simplify. Go from there. What do you get?