Solve the following equations analytically:

1) 8^((x^3)-x)=1

2) 45((1/5)^x)=10

Question

Solve the following equations analytically:

1)  8^((x^3)-x)=1

2)  45((1/5)^x)=10

anonymous · Answer

first one gives $x^3-x=0$ or $x(x^2-1)=0$ making $x(x+1)(x-1)=0$

anonymous · Answer

so $x=0$ or $x=1$ or $x=-1$

anonymous · Answer

so what happened to the 8 and the 1 in the first question

hero · Answer

^See, if you try to isolate x, you just end up eliminating it in the process.

anonymous · Answer

So what should the final answer be?

hero · Answer

Actually, take logs of both sides and see if that helps

hero · Answer

Yes, taking logs of both sides should work

hero · Answer

Because after taking logs of both sides, you'll end up with

$$(x^3 - x)\log(8) = \log(1)$$

hero · Answer

Then you'll have:

$$(x^3 - x) = \frac{\log(1)}{\log(8)}$$

hero · Answer

But log(1) = 0, so

$$x^3 - x = \frac{0}{\log(8)}$$

Thus:
$$x^3 - x = 0$$

hero · Answer

You should be able to solve from there

anonymous · Answer

ok i got -1, 0, and 1 for x

hero · Answer

Because $x^3 - x$ reduces to $x(x+1)(x-1)$

hero · Answer

Yes, you have it

anonymous · Answer

Thanks

anonymous · Answer

Do you know anything about #2

hero · Answer

Give me a minute

anonymous · Answer

ok

hero · Answer

1. Divide both sides by 45
2. Reduce 10/45
3. Take logs of both sides

anonymous · Answer

well, i guess i got it :) thanks for your help