Solve 5√x ≥ 625 Exponential Inequality

[enibaraliu]

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

5$$\sqrt[x]{64}$$ $$\geq$$ 625

5*2^6/x>=625 dived by 5
2^6/x>=125
2^6/x>=5³ and i have not idea what to do then

 

[symbolipoint]

Raise both sides to the x/6 power and obtain 2 >= 5^(3x/6). Then simplify and obtain
2 >= 5^(x/2)

From there the steps no longer seem clear unless you know how to use rules of logarithms. Using base 2 or base 5 appear to offer no advantage, so take logarithms of both sides, base 10. (or base e if you prefer).

 

[enibaraliu]

but the answer sad that the result is:xE (0,3),so this don't need logarithms

 

[symbolipoint]

Some of that solution is wrong. x cannot reasonably be too close to 3. Did you try using logarithms and resorting to a graphing calculator? x should be small, maybe very small. Anybody else? I might check more thoroughly, later.

 

[symbolipoint]

I believe easiest to start from here,
2^(6/x) >= 125
and take logarithms by any base of both sides.

 

[Tedjn]

There is no way I know of to get an exact answer for x without using logarithms. You can plug into a graphing calculator to get an approximate answer, but why not use logarithms?

 

[symbolipoint]

There is no way I know of to get an exact answer for x without using logarithms. You can plug into a graphing calculator to get an approximate answer, but why not use logarithms?

I in fact solved the problem without using a graphing calculator; I also then posted the solution process but doing so was a violation of the rules of this forum so that post was deleted. Knowing about logarithms is truly necessary.

 

[HallsofIvy]

2⁷= 128 so you must have 6/x slightly less than 7. that means that x is slightly less than 7/6, or close to 1.
I don't know what you meant by "but the answer said that the result is:xE (0,3),so this don't need logarithms " since many numbers in that interval can only be written as logarithms!