# derivates of the natutal logs of (constant \* x) **URL:** \ **Category:** Factual Questions\ **Created:** [October 26, 2010, 3:31am UTC](https://boards.straightdope.com/t/derivates-of-the-natutal-logs-of-constant-x/558378 "2010-10-26T03:31:49Z")\ **Posts on this page:** 5\ **Page:** 1
**Author:** ![Darth\_Panda](https://avatars.discourse-cdn.com/v4/letter/d/ee7513/32.png) [@Darth\_Panda](https://boards.straightdope.com/u/Darth_Panda)\ **Post date:** [October 26, 2010, 3:31am UTC](https://boards.straightdope.com/t/derivates-of-the-natutal-logs-of-constant-x/558378/1 "2010-10-26T03:31:49Z")
So, the derivate of ln(x) is 1/x. And the derivative of ln(2x) is also 1/x, and same for ln(3x), etc. So, since they all have the same derivative then either they are the same line, or parallel lines. Graphing indicates that they are parallel. The difference between the y values of f(x)=ln(x) and f(x)=ln(2x) for any x is 0.693; between ln(2x) and ln(3x) it’s 0.405; then 0.288, 2.223, and so forth. So, my question, where does the difference is values originate from -\> if the rate of change at any given x is identical between the two functions, where/when/how did the lines end up in different places? Most parallel lines involve the addition of a constant after manipulating the x term, but that isn’t the case here. So what’s going on? eta: could a mod fix the typo in the thread title? ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [October 26, 2010, 3:39am UTC](https://boards.straightdope.com/t/derivates-of-the-natutal-logs-of-constant-x/558378/2 "2010-10-26T03:39:15Z")
Logarithms turn multiplication into addition. E.g., ln(a \* b) = ln(a) + ln(b). In particular, ln(2x) = ln(x) + ln(2). ln(3x) = ln(x) + ln(3). And so on. So it _is_ just addition of a constant. Any two functions with the same derivative have a constant difference, as you yourself pointed out and even calculated, yet seemed to fail to fully appreciate. ---
**Author:** ![Darth\_Panda](https://avatars.discourse-cdn.com/v4/letter/d/ee7513/32.png) [@Darth\_Panda](https://boards.straightdope.com/u/Darth_Panda)\ **Post date:** [October 26, 2010, 3:49am UTC](https://boards.straightdope.com/t/derivates-of-the-natutal-logs-of-constant-x/558378/3 "2010-10-26T03:49:10Z")
> [@Indistinguishable](#): > > Logarithms turn multiplication into addition. E.g., ln(a \* b) = ln(a) + ln(b). > > In particular, ln(2x) = ln(x) + ln(2). ln(3x) = ln(x) + ln(3). And so on. So it _is_ just addition of a constant. Any two functions with the same derivative have a constant difference, as you yourself pointed out and even calculated, yet seemed to fail to fully appreciate. Sweet, thanks. Just didn’t put into context the ability to seperate out a constant. ---
**Author:** ![Ruminator](https://avatars.discourse-cdn.com/v4/letter/r/b9bd4f/32.png) [@Ruminator](https://boards.straightdope.com/u/Ruminator)\ **Post date:** [October 26, 2010, 4:08am UTC](https://boards.straightdope.com/t/derivates-of-the-natutal-logs-of-constant-x/558378/4 "2010-10-26T04:08:42Z")
> [@Darth\_Panda](#): > > Just didn’t put into context the ability to seperate out a constant. We could also use log base10 (with nice round numbers) to make the concept easier to remember. 1 million = 1,000,000 which is the same as multiplying 1,000 \* 1,000 or 10 \* 100,000. 10^3 + 10^3 = 1000000 log10(1000000) = 6 log10(1000 \* 1000) = 6 log10(1000) + log(1000) = 6 3 + 3 = 6 10^1 + 10^5 = 1000000 log10(10 \* 100000) = 6 log10(10) + log10(100000) = 6 1 + 5 = 6 Sometimes when I have issues thinking about ln(), I rethink it as log10() to work through the logic. Using log base10 is my mental crutch because my brain can’t calculate natural logs (2.71828 ^ ??? = ???) with arbitrary concrete numbers. ---
**Author:** ![Ruminator](https://avatars.discourse-cdn.com/v4/letter/r/b9bd4f/32.png) [@Ruminator](https://boards.straightdope.com/u/Ruminator)\ **Post date:** [October 26, 2010, 4:17am UTC](https://boards.straightdope.com/t/derivates-of-the-natutal-logs-of-constant-x/558378/5 "2010-10-26T04:17:52Z")
> [@Ruminator](#): > > 10^3 + 10^3 = 1000000 > > 10^1 + 10^5 = 1000000 Sorry, those should have been multiplication symbols instead of plus signs (but you probably saw passed that typo): 10^3 \* 10^3 = 1000000 10^1 \* 10^5 = 1000000 I just verified these with my big thick dusty book of logarithms and also my slide rule.