0
$\begingroup$

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?

$\endgroup$
2
$\begingroup$

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

$\endgroup$
2
$\begingroup$

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

$\endgroup$
0
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.