differentiation/integration with base other than e (1 Viewer)

[idling fire]

Could someone please give me the theory for this?
Just in case it's in the test tomorrow!!! eeek...

It's like:



but I think (2) might be wrong...

Help please.

 

[PC]

Isn't ∫ b dx just equal to bx + C?

1 and 3 appear to be right, but I had to work them through.

IMO, don't worry about trying to memorise all this stuff. Learn the change of base rule and be able to take the log of both sides of an equation. Then you can handle anything!

For example:
1. d/dx(log_ax)
= d/dx(ln x/ln a)
= 1/(ln a) . 1/x
= 1/(x ln a)

 

[Forbidden.]

1. It's correct, but I'd avoid using brackets on the answer.

2. Incorrect, the answer could only be correct if question was ∫b^x dx and neither did you add a constant (C) if it is an indefinite integral.

3. I know log_e x = ln x ... but I doubt log_b x = ln x let alone log_b x = x ln b ... change of base ?
I admit I don't know the correct answer to this either just at the top of my head.


Call in the Maths Team of airie, ianc, jyu, pLuvia or Riviet.


Anyway ...
I had to agree my Maths test was the best ever, it was pretty good, except there wasn't even a single exponential or logarithm function at all.
There was a Physics question too, who has heard of satellites traveling at approx. 2x10¹⁵ ms^-1 ? I hope a figure of around 7690 ms^-1 is more appropriate.

EDIT: Take a look at what PC has done for Q3, it may help.
EDIT2: ∫ b dx is equal to bx + C, as long as b is a variable not an exponential.

 

[pLuvia]

For 1)
b^x
=e^ln(bx)
=e^xln(b)
d/dx(b^x)=ln(b).e^xln(b)
=ln(b).b^x

2) This is just bx+C

And for 3) just look at PC's explanation

 

[idling fire]

Ok, thanks guys.

My mistake, (2) was supposed to be ∫ b^x

Now lets just hope I can remember things... :S

 

[Forbidden.]

Diffrentiation and Integration of other exponentials.

d(a^x)/dx = a^x ln a

∫a^x dx = (a^x / ln a) + C