Solve for x

1. Jul 7, 2011

nae99

1. The problem statement, all variables and given/known data

5^2x - 4(4^x) + 8 = 5

2. Relevant equations

3. The attempt at a solution

5^2x - 16^x + 8 = 5

5^2x - 16^x + 8 - 5 = 0

2. Jul 7, 2011

SammyS

Staff Emeritus
4(4x) = 41 4x = 4(x+1). (This is not the same as 16x.)

I suggest that you review properties of exponents.

It's difficult to work with logarithms (which are actually exponents) if you don't know how to work with exponents.

3. Jul 7, 2011

nae99

i dont properties of exponents

4. Jul 7, 2011

I like Serena

Let me check this.

You write:
5^2x - 4(4^x) + 8 = 5 ​

So did you mean:
$$5^2x - 4(4^x) + 8 = 5$$
or:
$$5^{2x} - 4(4^x) + 8 = 5$$
or yet something else?

Actually, I can't imagine you meant either the first or the second, because you won't be able to solve either.
You would only be able to solve the second form by trial and error, but I suspect that was not intended.

5. Jul 7, 2011

nae99

$$5^{2x} - 4(4^x) + 8 = 5$$

6. Jul 7, 2011

HallsofIvy

Staff Emeritus
On the other hand, if it were
$$4^{2x} - 4(4^x) + 8 = 5$$
It would be easy- using the "laws of exponents". If you do not know the laws of exponents ($a^x*a^y= a^{x+ y}$, $(a^x)^y= (a^y)^x= a^{xy}$), you should not be attempting a problem like this. Who ever gave you this problem clearly believes that you do know them. Learn the laws of exponents!

7. Jul 7, 2011

Ray Vickson

An obvious solution is x = 0. Numerical solution methods home in on this solution as well.

RGV

8. Jul 7, 2011

nae99

yes i am aware of those two laws but how will i apply it to:
$$5^{2x} - 4(4^x) + 8 = 5$$

9. Jul 7, 2011

I like Serena

You won't.
You'll only solve it numerically, but I do not think your current course is teaching you that.
(I still think you made a copying error when you typed in the problem. )

10. Jul 7, 2011

nae99

oh ok

11. Jul 7, 2011

nae99

no i did not make any error that is what i am seeing on the paper

12. Jul 7, 2011

I like Serena

Then the only way you'll solve it, is by trial and error (aka numerically).

Try filling in x=0, x=1, x=-1, x=2.
Make a graph.
Try and think of other values for x to try, like x=0.5.

There!

13. Jul 7, 2011

nae99

ok thanks

14. Jul 7, 2011

SammyS

Staff Emeritus
You can solve it graphically.

$5^{2x} - 4(4^x) + 8 = 5$ is equivalent to $25^{x} - 4(4^x) + 3 = 0$

Graph $y= 25^{x} - 4(4^x) + 3$ and find the y-intercepts.

15. Jul 7, 2011

nae99

ok i will try that

16. Jul 7, 2011

HallsofIvy

Staff Emeritus
I will suggest, once again, that you look at the problem again and make sure it is not
$$4^{2x}−4(4^x)+8=5$$
which, as I said before, would be easy.

17. Jul 7, 2011

nae99

yep, its not 4 it is 5

18. Jul 7, 2011

SammyS

Staff Emeritus
Hello, HoI, (Check for a PM)

While you are correct that $4^{2x}−4(4^x)+8=5$ would be a more reasonable problem to solve, the solutions to $5^{2x}−4(4^x)+8=5$ are rational, which is surprising to me.

Last edited: Jul 7, 2011
19. Jul 7, 2011

nae99

ok then, thanks

20. Jul 7, 2011

spamiam

I'm not sure if this works, and it's definitely beyond the scope of precalculus, but we could try to use the generalized binomial theorem to look for rational solutions at least:

\begin{align*} 0 &= 5^{2x} - 4 \cdot 4^x + 3 = (1+4)^{2x} - 4^{x+1} + 3 = \sum_{k=0}^\infty \binom{2x}{k} 4^k - 4^{x+1} + 3 = \sum_{k=1}^\infty \binom{2x}{k} 4^k - 4^{x+1} + 4 \\ &= 4 \left(\sum_{k=1}^\infty \binom{2x}{k} 4^{k-1} - 4^x + 1 \right). \end{align*}

(Strictly speaking the infinite series might diverge, but I think it might converge 2-adically.) But I'm not sure how to proceed now... Expand $4^x$ as a power series?