# Help on logarithmic differentiation problem

1. Jan 23, 2009

### crm08

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

Use logarithmic differentiation to find the derivative of the function.

2. Relevant equations

y = (sin(x))^(ln(x))

3. The attempt at a solution

I guessing the first step is raise both sides to "e", but so far I have only completed problems by taking the natural log of both sides.

(1) e^(y) = e^(sin(x))^(ln(x))

(2) d/dx (e^y) = d/dx (e^(sin(x))^(ln(x)))

(3) Can I start moving exponents in front of "e" to use the product rule, if so, do I bring "sin(x)^(ln(x)) down?, or am I on the completely wrong track?

2. Jan 23, 2009

### tiny-tim

Hi crm08!
= e^(sin(x)*ln(x))

3. Jan 23, 2009

### crm08

Oh I gotcha, so now use:

d/dx (e^g(x)) = e^g(x) * g'(x), where g(x) = (sin(x))(ln(x)), and use product rule for g'(x)

I think I'm on the right track

4. Jan 23, 2009

### wsalem

use ln not e, since $$\ln{x^a} =a \ln{x}$$
So that $$\ln{y} = \ln( \sin{x}^{\ln{x}} ) = \ln(x) \ln(sin{x})$$

5. Jan 23, 2009

### wsalem

tiny-tim, I'm confused, why would $$e^y = e^{(\sin(x)^{\ln(x)})}$$ equals $$e^{(\sin(x)*\ln(x))}$$

6. Jan 23, 2009

### crm08

ok i'm getting

(1/y) * dy/dx = (ln(x))*(1/sin(x))*(cos(x)) + (ln(sin(x))*(1/x)

= y*[(ln(x))*(tan(x)) + ((ln(sin(x))) / x)]

am I getting any closer?

7. Jan 23, 2009

### Gib Z

It wouldn't. In general, (a^b)^c = a^(bc), but not a^(b^c). Take an example using a=3, b=2, c=3.

(a^b)^c = 729= a^(bc). a^(b^c) = 3^8, different.

I'm sure tiny-tim just got a bit confused.

As for crm08 - the tan should be a cot, and you should not use an equals signs when you actually multiplied by y, but other than that its looking sweet.

8. Jan 23, 2009

### tiny-tim

oops!

mmm … all those ^s

it'll be a lot easier if everyone uses the X2 tag (just above the reply box)

let's see, it should have been …

sinxlnx = (eln(sinx))lnx = eln(sinx)*lnx

9. Jan 23, 2009

### wsalem

ahhh, so you were referring to y and not $$e^y$$ that equals $$e^{\ln(sinx)*\ln{x}}$$.
That makes sense now!