# Solving Logarithmic Inequality

Solve $$\log$$ $$\left(5^{\frac{1}{x}}+5^{3}\right)<\log$$ $$6$$ $$+$$ $$\log$$ $$5^{\left(1+\frac{1}{2x}\right)}$$

It is possible to simplify the inequality using the quotient rule property of logarithms, and disregard the logarithms, considering same bases, into: $$\left(\frac{5^{\frac{1}{x}}+5^{3}}{5^{\left(1+\frac{1}{2x}\right)}}\right)<6$$

Is this a step in the right direction? And, how are we able to isolate the variable in such an inequality?

$$5^{1+\frac{1}{2x}}=5\times (5^{1/x})^{1/2}$$
Position $$(5^{1/x})^{1/2}=t$$
Hint: Use the substitution $$t=5^{\frac{1}{2x}}$$ and the inequality become: $$\frac{t^2+125}{5t}<6$$
Using your idea we get $$5^{1/x}+5^3<30\cdot (5^{1/x})^{1/2}$$ let $$t=5^{1/x}$$ so you have to solve $$t+125<30\sqrt{t}$$