# Thread: what to do then?

1. ## what to do then?

I tried to solve this exponential inequation , but i can't do more:

5 625

5*2^6/x>=625 dived by 5
2^6/x>=125
2^6/x>=5^3 and i have not idea what to do then

2. Originally Posted by MathE1
I tried to solve this exponential inequation , but i can't do more:

5 625

5*2^6/x>=625 dived by 5
2^6/x>=125
2^6/x>=5^3 and i have not idea what to do then
1. Multiply by x (according to the first line x > 0):

$2^6 \geq 5^3 \cdot x~\implies~x\leq\frac{2^6}{5^3}$ and $\frac{2^6}{5^3}= \frac{4^3}{5^3} = \left(\frac45\right)^3 = 0.512$

3. Originally Posted by earboth
1. Multiply by x (according to the first line x > 0):

$2^6 \geq 5^3 \cdot x~\implies~x\leq\frac{2^6}{5^3}$ and $\frac{2^6}{5^3}= \frac{4^3}{5^3} = \left(\frac45\right)^3 = 0.512$
thanks earboth but i did one mistake
it's 2^(6/X)>=5^3

2^(6/X)>=5^3

5. Originally Posted by MathE1
$2^{\frac6x} \geq 5^3$
Take the inequality to the power of $\frac x6$:
$\left(2^{\frac6x}\right)^{\frac x6} \geq \left(5^3\right)^{\frac x6}~\implies~2 \geq 5^{\frac x2}$ Now square the both sides of the inequality:
$4 \geq 5^x~\implies~\log_5(4) \geq x$
$x\leq \log_5(4)$ and $\log_5(4)=\dfrac{\ln(4)}{\ln(5)}\approx 0.861...$