So what I want to know is what happened below, when the derivative was taken and the logs went away. I know that it’s lnb/lna, and the lna is ln10 or log base 10, but where did it go? Did it get zeroed out, as one does in taking the derivative of a constant? Where did ln10 go? What happened there?

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If I understand your question correctly, you need to apply the second formula from step 2 for taking the first derivative of the log function. This makes the log “disappear”. Ofcourse you will also have to apply the chain rule.

OK, but that’s ln and what I’m given is log. log subscript a b = lnx/lna, and ln of logab is ln10.

The transition from log to ln is what I don’t understand.

I’m not quite sure, but they may have gotten sloppy and used log and ln interchangeably. Not that it matters.

log(x) = ln(x) / ln(10)

If both sides have been divided by ln(10), it comes out in the wash.

Wow, thanks that helps a lot Mr. Rational!

d/dx (log y) = d/dx (ln y / ln 10)

Factor out the coefficient, (1/ln 10) d/dx (ln y) = (1/ln 10) * 1/y * dy/dx

Similarly for the RHS. You’d have everything stated times 1/(ln10) on both sides, which would cancel.

The author surely intended to use natural logs but got sloppy.

OK, now that makes sense. thanks!

The author didn’t get sloppy, he wants to make it more difficult and messy for the AP Calc high school teenyboppers. This was my daughter’s homework question. I never saw anything like it at State U Calculus 1 class.

How should the problem have been presented to make it more readable/understandable?

showing the ln10 as reciprocals and showing them cancel out

In mathematics “log” generally means the the natural log (log base e). More advanced math texts will rarely use the ln notation. In this case you can figure it out from context that log is being used to mean log base e rather than log base 10.

Since the function was applied to both sides of the equation it was the author’s choice of which function to apply – since the notation was ambiguous (using just “log”) you can assume he used the one that didn’t require him to take the extra step.

Another line in step five to show all the ln10s?

yeah, I’d say. Are you an AP calculus teacher or math book writer?

Neither, I was curious about what would be considered something that wouldn’t have to be a step, and what would be.

The set should have used ln throughout, for consistency. Since the differentiation operation would have been affected this counts as an error.

Don’t be afraid of all the symbols, MW. Even a monster like that can be broken down into basic steps.

On a clear day, you are on an aeroplane which is at 38,000 ft above the middle of the Pacific ocean. Taking the radius of the earth as 6,400km, what is the approximate distance between you and the horizon of the earth? (1ft=0.3048m)

130km

390km

700km

1,300km

390 km.

You just made me waste time at work! Stop that!