Differentiate logarithmic function - how are they getting e?: 10^(2 sqrt(t))

[Strat]

Problem: 10^(2 sqrt(t))
Answer: r'(t) = [(ln10)10^(2 sqrt t) / sqrt(t)

How did they magic a natural log out of that? I must be forgetting some log rule. I thought ln always had base e and I see no e in the problem so I am confused.

 

[stapel]

Problem: 10^(2 sqrt(t))
Answer: r'(t) = [(ln10)10^(2 sqrt t) / sqrt(t)

How did they magic a natural log out of that? I must be forgetting some log rule. I thought ln always had base e and I see no e in the problem so I am confused.

If all they gave you is what is in the posted "Problem", then I have no idea how "they" "got" anything. For instance, how does "r'(t)" relate to the "Problem"? How did you know what to do, lacking any instructions? Perhaps if there were more to this "Problem", we might be able to advise.

 

[HallsofIvy]

Problem: 10^(2 sqrt(t))
Answer: r'(t) = [(ln10)10^(2 sqrt t) / sqrt(t)

How did they magic a natural log out of that? I must be forgetting some log rule. I thought ln always had base e and I see no e in the problem so I am confused.

The problem is to differentiate a power of x and I suspect that you do not know a "rule" for differentiating powers of 10! You do, I am sure, know how differentiate a power of e.

So convert one to the other. The "log rule" you want is really the fact that \(\displaystyle e^x\) and ln are inverse functions to one another: \(\displaystyle e^{ln(x)}= x\). From that, \(\displaystyle 10^x= e^{ln(10^x)}= e^{x ln(10)}\).

Now differentiate that, using the chain rule. \(\displaystyle \frac{de^u}{dt}= e^u\frac{du}{dt}\). Here, \(\displaystyle u= 2\sqrt{t}= 2t^{1/2}\) so that \(\displaystyle \frac{du}{dx}= 2(1/2)t^{-1/2}\).

So the derivative is \(\displaystyle t^{-1/2}e^{2\sqrt{t}}\).

 

[Strat]

The problem is to differentiate a power of x and I suspect that you do not know a "rule" for differentiating powers of 10! You do, I am sure, know how differentiate a power of e.

So convert one to the other. The "log rule" you want is really the fact that \(\displaystyle e^x\) and ln are inverse functions to one another: \(\displaystyle e^{ln(x)}= x\). From that, \(\displaystyle 10^x= e^{ln(10^x)}= e^{x ln(10)}\).

Now differentiate that, using the chain rule. \(\displaystyle \frac{de^u}{dt}= e^u\frac{du}{dt}\). Here, \(\displaystyle u= 2\sqrt{t}= 2t^{1/2}\) so that \(\displaystyle \frac{du}{dx}= 2(1/2)t^{-1/2}\).

So the derivative is \(\displaystyle t^{-1/2}e^{2\sqrt{t}}\).

Thanks