Question

Solve: 64^(x – 3) = 42x

I know that with most exp equations, the exponent is on one side and just a number is on the other. I'm unsure about this one since there's that x next to the 42.

Answer 1

It was actually someone else on here who asked this and I couldn't answer it b/c I didn't know what to do with the x on the right side :-/

Answer 2

and I'm so nerdy that I just have to know haha

Answer 3

hold on a sec please. i will brb

Answer 4

Sorry, im back.

Answer 5

wb

Answer 6

i know how to do this, just did this years ago. Let me recall.

Answer 7

Lol I have the same feeling about this thing!

Answer 8

I got as far as 64^(x-3)/42 = x
and then
(x-3)log(64/42) = log(x)

Answer 9

i think i found the way out, but i dont know how to write it,

How can we separate 64^(x – 3) ?

Answer 10

sec I'll look up exponential properties

Answer 11

Sure

Answer 12

http://www.efunda.com/math/exp_log/exp_relation.cfm

Answer 13

so I guess... 64^x/64^3 ?

Answer 14

i will refer to my register. let me chk

Answer 15

Asked for help.

Answer 16

If it's too much trouble then that's okay. Who knows, maybe the person who originally asked the question typed it out wrong :-?

Answer 17

Thank you for working at it though :-)

Answer 18

Lol, it's not wrong. that's for sure.

Answer 19

Well I've gotta go read linear algebra (which I don't want to do), so have a nice evening!

Answer 20

I'll check back in a little bit.

Answer 21

i will send u the solution.. i want to learn this as well

Answer 22

hahaha I've gotta stop getting people all curious!

Answer 23

ttyl

Answer 24

Lol, that's a no problem.

Answer 25

I think its 64^(x – 3) = 42^x

Answer 26

Oh damn that would make it so much easier

Answer 27

(x-3)log(64) = xlog(42)

(x-3)/x = log(42)/log(64)

1-3/x = log(42)/log(64)

-x/3 = log(64)/log(42) -1

And so on and so-forth, I'm too lazy to do the rest :-)

Answer 28

what is the level of the question i mean 2 say which class ??

Answer 29

Another user asked it and I was curious. It's precalc level I believe.

Answer 30

graphical solution can be applied or
by taking log and then using hit and trial method or
u can get the answer using scientific calculator which uses trial method for range of values provided

Answer 31

is -1/log(64) the answer?

Answer 32

u have taken out x common which wrong...... xln64-ln(42x) !=x(ln64-ln42)

Answer 33

-1/log(64) is not correct.
3 log(64) / (log(64) - log(42)) is the correct for the equation
\[63^{x-3}=42^{x}\]
and incorrect for the equation
\[64^{x-3}=42x\]

The truth is there is no analytic answer. There is only a numerical solution to this kind
of equation (transcendental equation).
Like swarup169 said you can solve it a few ways:
graphically, with a good calculator, Newton's method, iteration, Taylor series,...
Graphically you would plot both sides of the equation then find the x components of
the intersections.

By means of iteration we solve the equation for only one of the two x-s.
\[64^{x-3}=42x\]
\[x=\frac{64^{x-3}}{42}\]
Then we make a guess of the answer of the original equation. For example x=1.
This obviously is not the correct answer but it is probably close to it. So we plug this x1=1
into the equation and we will get a better approximation for the correct value of x.
\[x2=\frac{64^{x1-3}}{42}=\frac{64^{1-3}}{42}=\frac{64^{-2}}{42}=\frac{1}{42*64^{2}}\approx1.52\]
Now we have x2 and again we put it back into the equation:
\[x3=\frac{64^{x2-3}}{42}=\frac{64^{1.52-3}}{42}=\frac{1}{42*64^{1.48}}\approx5*10^{-5}\]
Keep doing this untill x does not change drastically.
\[x4=\frac{64^{x3-3}}{42}\approx9.1*10^{-8}\]
\[x5=\frac{64^{x4-3}}{42}\approx9.0826^{-8}\]
Et cetera untill you have the desired accuracy.

This equation actually has two real solutions. I noticed this when I plotted both sides
of the equation. So to get the other solution use the same method, only solve for the x on
the other side:
\[x=\frac{\ln(42x)}{\ln(64)}+3\]
Guess the answer like x1=4.
\[x2=\frac{\ln(42x1)}{\ln(64)}+3\approx4.23\]
\[x3=\frac{\ln(42x2)}{\ln(64)}+3\approx4.25\]
\[x3=\frac{\ln(42x2)}{\ln(64)}+3\approx4.247\]
Et cetera...

So the two solutions are
\[x \approx9.0826*10^{-8}\]
\[x \approx4.247\]