How do I solve exponential equations? Ex) 32^x=16^(x+1) Sorry to bug you guys just confused w. some of my work

Question

anonymous · Answer

We can start by using the property of logs.
For example,
$a^x \implies x \ln(a)$
This brings the variable out of the exponent

anonymous · Answer

Do I do this for each side?

anonymous · Answer

xln32 and x+1ln16

anonymous · Answer

Yes. Whenever you take the log/natural log, you have to do it to both sides.

anonymous · Answer

Okay, now try to isolate the x's. You'll have to divide over the ln(16) from the right and then divide the x from the left.

anonymous · Answer

Awesome! Right now I have xln32=x+1ln16 So I am going to divide 16 on each side

anonymous · Answer

No -- you have to divide by the natural log of 16, ln(16)

anonymous · Answer

I have (xln2)=x+1?

anonymous · Answer

I divided 32/16 and the 16 canceled out on the right

anonymous · Answer

Sorry if I am not understand you correctly.

anonymous · Answer

Aha! You fell for one of the biggest and most common mistakes when dividing by logs. Do you have a calculator? Input these two into your calculator:
$$(1)~~~~~\frac{ \ln(32) }{ \ln(16) }$$
$$(2)~~~~~\ln (\frac{ 32 }{ 16 })$$

anonymous · Answer

1. I got 1.25 or 5/4 2. I got .693

anonymous · Answer

So we see that they are not the same, therefore we cannot do this simplification that you did. Why is that? Let's deviate from our equation for a second and look at properties of logs.

`Property of Subraction:`$$\ln(a)-\ln(b)=\ln(\frac{ a }{ b })$$As we can see here, in order to simplify by division, we have to subtract the two full log terms. But we're not subtracting in our problem, we're dividing. So we must take the full value of ln(32) divided by the full value of ln(16). We cannot simplify this any further.

freckles · Answer

32^x=16^(x+1) 
you can also do this without logs 
(not all can be done without logs but this one can be)
since 32=2^5 and 16=2^4

anonymous · Answer

True! I think it's always best to learn it with logs but it is also good to know shortcuts when they can be applied. Work smarter not harder! X)

anonymous · Answer

I remember learning it without logs.

anonymous · Answer

But guys I really have to run out really quick. I'll be back thanks Cshrix & Freckles!!

anonymous · Answer

i'll be back in 20

anonymous · Answer

i'll be back in 20