I need a bit of help rearranging this equation

[Hfuais]

1. The equation is : y = 50 ( 5 - 0.67 ^ x)



2. Just basic rearranging, I need to isolate x.



3. So far I have..
y = 50 ( 5 - 0.67 ^ x)
y - 50=( 5 - 0.67 ^ x)
y - 55 = - 0.67 ^ x)
Now is the bit where I am stuck, seeing it is raised as a power? I know it's simple, I just can't remember! :/

 

[dystopia]

Remember what the natural log (ln) is and this becomes simple.

$55 - y = .67^x$

$ln(55 - y) = ln(.67^x)$

$ln(55 - y) = xln(.67)$

$x = ln(55 - y) / ln(.67)$

Is this what you needed?

 

[HallsofIvy]

You are doing pretty much everything wrong. To begin with, you do not just "move to the other side and change the sign". I wish that phrase we banned!

To "solve for x" or "make x the subject" (the first is American English, the second British English) you need to undo everything that is done to x, by doing the opposite, on both sides.

The first thing you want to eliminate is that "50" and it multiplies the rest of the right side and the opposite of multiplying is dividing. Divide both sides by 50 to get $y/50= 5- 0.67^x$. Now, we have 5 added on the right so subtract 5 from both sides: $y/50- 5= -0.67^x$. That "-" indicates a multiplication by -1 so divide by -1 (which, yes, is the same as multiplying by -1): $5- y/50= 0.67^x$. Note that 5- y/50 is NOT (5- y)/50. We could write 5= 250/50 so that 5- y/50= (250- y)/50 so that $5- y/50= (250- y)/50= 0.67^x$.

Now the right side is a exponential function- the "x" is in the exponent. And the opposite of an exponential is a logarithm. That is, take the logarithm of both sides:
log((250- y)/50)= log(.67^x)= xlog(.67)

Finally, since we now have the right side reduced to x times something, divide by it: x= [log((250- y)/50)/log(.67).