Solving Logarithmic Inequality

Question

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?

Answer 1

Hint

$$5^{1+\frac{1}{2x}}=5\times (5^{1/x})^{1/2}$$

Position $(5^{1/x})^{1/2}=t$

Then noting that the exponential is always positive you can multiply my the denominator both sides and avoid a quotient disequation

Answer 2

Hint: Use the substitution $t=5^{\frac{1}{2x}}$ and the inequality become: $$\frac{t^2+125}{5t}<6$$

Answer 3

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 $$t+125<30\sqrt{t}$$