Tough logarithmic equation

**Arka** · May 11th 2011, 08:42 AM

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.

Please Help.

Thanks in advance

**red_dog** · May 11th 2011, 09:04 AM

$\log_5\left(5^{\frac{1}{x}}+125\right)=\log_56+ \log_5 \left(5^{1+\frac{1}{2x}}\right)$

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

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

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

Can you continue?

**Ackbeet** · May 11th 2011, 09:08 AM

Write the RHS all as log base five quantities. Then the RHS will simplify. Go from there. What do you get?

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