Proving that two Ln Equations equal each other

[Liparulo]

Homework Statement

y=8^0.01x
Differentiates to: 0.01 ln8 x 8^0.01x
Using derive on a graphics calculator, the answer given is:
3x2^3x-200/100 ln2/25
Prove that the two equal each other using index and logarithm laws.

Homework Equations

The Attempt at a Solution

Ah, well, I've had plenty of attempts. I've no idea if any are on the right track, but I'm working backwards from the second equation and have tried to split the equation into three parts to simplify. Needless to say, it didn't work. I've also tried simplifying the top of the equation to no avail. Any help would be appreciated.

 

[Mark44]

Homework Statement

y=8^0.01x
Differentiates to: 0.01 ln8 x 8^0.01x
Using derive on a graphics calculator, the answer given is:
3x2^3x-200/100 ln2/25

Please use parentheses on the expression above, as it is very ambiguous. Also, do not use 'x' to indicate multiplication since x is a variable in this expression.

Prove that the two equal each other using index and logarithm laws.

Homework Equations

The Attempt at a Solution

Ah, well, I've had plenty of attempts. I've no idea if any are on the right track, but I'm working backwards from the second equation and have tried to split the equation into three parts to simplify. Needless to say, it didn't work. I've also tried simplifying the top of the equation to no avail. Any help would be appreciated.

 

[Liparulo]

I've done redone it in Microsoft word:
http://img175.imageshack.us/img175/6467/78807985.jpg

Thank you. :)

 

[Tedjn]

What happened to the 2^(-200/100)? What is 3*ln(2)?

 

[Liparulo]

I'm probably wrong, but I went:
2^{(3x/100) - 2}
2^(3x)0.01
8^0.01x

 

[Tedjn]

But in fact 2^(3x/100)-2 = 2^(3x/100)/2².

 

[Liparulo]

I'm not sure, I just need to prove that the two equations are equal. :|

 

[Tedjn]

Yes, if you look carefully, you'll see that your original simplification was wrong, because it forgot the 2² in the denominator. After you fix that, you can simplify further by considering how to condense 3*ln(2).

 

[Liparulo]

I'm sorry for sounding dumb, but could you please explain why 2² needs be in the denominator? Does not seem to be a good maths day today.

 

[Tedjn]

Before you went from 2^{(3x/100) - 2} to 2^(3x)0.01. These two are not equal, because in the first you are subtracting 2 from the exponent. In other words, the second expression is 2² times bigger than the first.

 

[Liparulo]

Hmm, okay then. That did occur to me initially, but I overlooked it because it started to match the other equation. How would you solve it? I've tried a few times to no avail.

 

[Tedjn]

Changing to 8^0.01x is fine, as long as you remember to divide the whole expression by 2². This should get you closer. Next, simplify 3*ln(2).

 

[Mark44]

The result from the calculator was
$$\frac{3 ln 2 * 2^{(3x - 200)/100}}{25}$$
$$= \frac{ln(2^3) * 2 ^{.03x - 2}}{25}$$
$$= \frac{ln(8) * 2^{.03x} * 2^{-2}}{25}$$

I used the fact that a^{b - c} = a^b * a^-c.
Another property of exponents that will be useful for finishing this is a^bc = (a^b)^c.

 

[Liparulo]

Ah, I solved it! Thank you very much.