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The definition of $ln(x)$
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When I taught my student the logarithm, he asked me about the historical definition of $ln(x)$.
- The first definition I found is that $$ln(x)=int_1^x fracdtt $$
- Defined as the logarithm to base $e$ or the inverse function of the exponentiation to base
$e$: $$ln(x)=y Longleftrightarrow e^y=x$$ where $e$ defined as
$$e=lim_ntoinftyleft( 1+frac1n right)^n$$
Which is the real definition of the logarithm?
calculus logarithms math-history
edited Jul 29 at 20:45
Xander Henderson
13.1k83150
asked Jul 29 at 20:42
El Mouden
589
 |Â
show 7 more comments
up vote
0
down vote
favorite
When I taught my student the logarithm, he asked me about the historical definition of $ln(x)$.
- The first definition I found is that $$ln(x)=int_1^x fracdtt $$
- Defined as the logarithm to base $e$ or the inverse function of the exponentiation to base
$e$: $$ln(x)=y Longleftrightarrow e^y=x$$ where $e$ defined as
$$e=lim_ntoinftyleft( 1+frac1n right)^n$$
Which is the real definition of the logarithm?
calculus logarithms math-history
edited Jul 29 at 20:45
Xander Henderson
13.1k83150
asked Jul 29 at 20:42
El Mouden
589
9
en.wikipedia.org/wiki/History_of_logarithms
– vadim123
Jul 29 at 20:45
2
Neither is the "real" definition of the natural logarithm. Or both are. They are equivalent---you can start with either and get to the other (and there are other definitions, too, e.g. in terms of a power series). Historically, it is likely that neither is the first definition to appear, either (Napier's tables are probably where logarithms get their modern start). Pedagogically, pick the one that works best for your students, and which best matches with the background they have.
– Xander Henderson
Jul 29 at 20:50
1
@XanderHenderson yeah i know , but my question is which of them is the first historical definition
– El Mouden
Jul 29 at 20:53
2
Please edit your question rather than just clarifying it in a comment.
– Rob Arthan
Jul 29 at 20:58
2
See hsm.stackexchange.com/questions/42/…
– quid♦
Jul 29 at 21:00
 |Â
show 7 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
When I taught my student the logarithm, he asked me about the historical definition of $ln(x)$.
- The first definition I found is that $$ln(x)=int_1^x fracdtt $$
- Defined as the logarithm to base $e$ or the inverse function of the exponentiation to base
$e$: $$ln(x)=y Longleftrightarrow e^y=x$$ where $e$ defined as
$$e=lim_ntoinftyleft( 1+frac1n right)^n$$
Which is the real definition of the logarithm?
calculus logarithms math-history
edited Jul 29 at 20:45
Xander Henderson
13.1k83150
asked Jul 29 at 20:42
El Mouden
589
When I taught my student the logarithm, he asked me about the historical definition of $ln(x)$.
- The first definition I found is that $$ln(x)=int_1^x fracdtt $$
- Defined as the logarithm to base $e$ or the inverse function of the exponentiation to base
$e$: $$ln(x)=y Longleftrightarrow e^y=x$$ where $e$ defined as
$$e=lim_ntoinftyleft( 1+frac1n right)^n$$
Which is the real definition of the logarithm?
calculus logarithms math-history
edited Jul 29 at 20:45
Xander Henderson
13.1k83150
asked Jul 29 at 20:42
El Mouden
589
edited Jul 29 at 20:45
Xander Henderson
13.1k83150
edited Jul 29 at 20:45
Xander Henderson
13.1k83150
edited Jul 29 at 20:45
Xander Henderson
13.1k83150
13.1k83150
asked Jul 29 at 20:42
El Mouden
589
asked Jul 29 at 20:42
El Mouden
589
asked Jul 29 at 20:42
El Mouden
589
589
9
en.wikipedia.org/wiki/History_of_logarithms
– vadim123
Jul 29 at 20:45
2
Neither is the "real" definition of the natural logarithm. Or both are. They are equivalent---you can start with either and get to the other (and there are other definitions, too, e.g. in terms of a power series). Historically, it is likely that neither is the first definition to appear, either (Napier's tables are probably where logarithms get their modern start). Pedagogically, pick the one that works best for your students, and which best matches with the background they have.
– Xander Henderson
Jul 29 at 20:50
1
@XanderHenderson yeah i know , but my question is which of them is the first historical definition
– El Mouden
Jul 29 at 20:53
2
Please edit your question rather than just clarifying it in a comment.
– Rob Arthan
Jul 29 at 20:58
2
See hsm.stackexchange.com/questions/42/…
– quid♦
Jul 29 at 21:00
 |Â
show 7 more comments
9
en.wikipedia.org/wiki/History_of_logarithms
– vadim123
Jul 29 at 20:45
2
Neither is the "real" definition of the natural logarithm. Or both are. They are equivalent---you can start with either and get to the other (and there are other definitions, too, e.g. in terms of a power series). Historically, it is likely that neither is the first definition to appear, either (Napier's tables are probably where logarithms get their modern start). Pedagogically, pick the one that works best for your students, and which best matches with the background they have.
– Xander Henderson
Jul 29 at 20:50
1
@XanderHenderson yeah i know , but my question is which of them is the first historical definition
– El Mouden
Jul 29 at 20:53
2
Please edit your question rather than just clarifying it in a comment.
– Rob Arthan
Jul 29 at 20:58
2
See hsm.stackexchange.com/questions/42/…
– quid♦
Jul 29 at 21:00
9
9
en.wikipedia.org/wiki/History_of_logarithms
– vadim123
Jul 29 at 20:45
en.wikipedia.org/wiki/History_of_logarithms
– vadim123
Jul 29 at 20:45
2
2
Neither is the "real" definition of the natural logarithm. Or both are. They are equivalent---you can start with either and get to the other (and there are other definitions, too, e.g. in terms of a power series). Historically, it is likely that neither is the first definition to appear, either (Napier's tables are probably where logarithms get their modern start). Pedagogically, pick the one that works best for your students, and which best matches with the background they have.
– Xander Henderson
Jul 29 at 20:50
Neither is the "real" definition of the natural logarithm. Or both are. They are equivalent---you can start with either and get to the other (and there are other definitions, too, e.g. in terms of a power series). Historically, it is likely that neither is the first definition to appear, either (Napier's tables are probably where logarithms get their modern start). Pedagogically, pick the one that works best for your students, and which best matches with the background they have.
– Xander Henderson
Jul 29 at 20:50
1
1
@XanderHenderson yeah i know , but my question is which of them is the first historical definition
– El Mouden
Jul 29 at 20:53
@XanderHenderson yeah i know , but my question is which of them is the first historical definition
– El Mouden
Jul 29 at 20:53
2
2
Please edit your question rather than just clarifying it in a comment.
– Rob Arthan
Jul 29 at 20:58
Please edit your question rather than just clarifying it in a comment.
– Rob Arthan
Jul 29 at 20:58
2
2
See hsm.stackexchange.com/questions/42/…
– quid♦
Jul 29 at 21:00
See hsm.stackexchange.com/questions/42/…
– quid♦
Jul 29 at 21:00
 |Â
show 7 more comments
2 Answers
2
active
oldest
votes
up vote
1
down vote
I'm no expert on maths history, but logarithms are old enough not to have a "historical definition" that meets our standards of what a definition should be. I think the integral definition of the logarithm is the better one to teach, for a few reasons:
What is exponentiation? Even if you define $e$, that may instantly tell you $e^2$ or $e^-frac13$, but how do you tell what $e^pi$ is? No combination of repeated multiplication, inversion, or taking roots of $e$ will produce this number. (Note, this can be mediated by defining $exp(x) = lim_ntoinfty left(1 + fracxnright)^n$).
The indefinite integral of $frac1x$ is such a natural question that it warrants the invention of a function to fill the gap.
The log laws (and hence exponential laws) turn into lovely applications of various integral rules.
The calculus properties of $ln$ and $exp$ follow immediately from this definition too.
That's why I would teach the integral definition.
answered Jul 29 at 21:00
Theo Bendit
11.8k1841
add a comment |Â
up vote
0
down vote
The fact that they call it a "logarithm" implies the must have had a concept that it is the logarithm of some base. So when the defined they must have been using the concept $ln x = y iff e^y = x$. And I even imagine they would be aware that $frac db^xdx = C_b*b^x$ (for rational values of $x$; irrational values would have been poorly understood) so that would figure there must be a base so that $C_b = 1$ and $frac de^xdx = e^x$.
But although that can be the concept and germination of a definition, it can't actually be a practical definition until after they had some way of finding what $e$ would be. And I imagine to do that they had to recognize that $int frac 1x dx$ is a logrithmic function and the value of it's base would be $lim (1 +frac 1n)^n$.
So I would guess, it went in this order 1) $frac db^xdx C(b)*b^x$ for some function $C(b)=limfrac b^h - 1h$. 2) That therefore $C(b) = int_1^b frac 1t dt$ and that $C(b)$ 3) the $e$ so that $C(b) = 1$ is $lim(1 + frac 1n)^n$ then 4) noting $log_e (x) = C(x)$ is an immediate consequence and then the final definition 4) $ln x := log_e x = int_1^xfrac 1t dt = C(x)$.
But I'm just guessing.
answered Jul 29 at 22:07
fleablood
60.2k22575
Guessing isn't very useful for determining the actual history...
– Hans Lundmark
Jul 30 at 7:40
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
I'm no expert on maths history, but logarithms are old enough not to have a "historical definition" that meets our standards of what a definition should be. I think the integral definition of the logarithm is the better one to teach, for a few reasons:
What is exponentiation? Even if you define $e$, that may instantly tell you $e^2$ or $e^-frac13$, but how do you tell what $e^pi$ is? No combination of repeated multiplication, inversion, or taking roots of $e$ will produce this number. (Note, this can be mediated by defining $exp(x) = lim_ntoinfty left(1 + fracxnright)^n$).
The indefinite integral of $frac1x$ is such a natural question that it warrants the invention of a function to fill the gap.
The log laws (and hence exponential laws) turn into lovely applications of various integral rules.
The calculus properties of $ln$ and $exp$ follow immediately from this definition too.
That's why I would teach the integral definition.
answered Jul 29 at 21:00
Theo Bendit
11.8k1841
add a comment |Â
up vote
1
down vote
I'm no expert on maths history, but logarithms are old enough not to have a "historical definition" that meets our standards of what a definition should be. I think the integral definition of the logarithm is the better one to teach, for a few reasons:
What is exponentiation? Even if you define $e$, that may instantly tell you $e^2$ or $e^-frac13$, but how do you tell what $e^pi$ is? No combination of repeated multiplication, inversion, or taking roots of $e$ will produce this number. (Note, this can be mediated by defining $exp(x) = lim_ntoinfty left(1 + fracxnright)^n$).
The indefinite integral of $frac1x$ is such a natural question that it warrants the invention of a function to fill the gap.
The log laws (and hence exponential laws) turn into lovely applications of various integral rules.
The calculus properties of $ln$ and $exp$ follow immediately from this definition too.
That's why I would teach the integral definition.
answered Jul 29 at 21:00
Theo Bendit
11.8k1841
add a comment |Â
up vote
1
down vote
up vote
1
down vote
I'm no expert on maths history, but logarithms are old enough not to have a "historical definition" that meets our standards of what a definition should be. I think the integral definition of the logarithm is the better one to teach, for a few reasons:
What is exponentiation? Even if you define $e$, that may instantly tell you $e^2$ or $e^-frac13$, but how do you tell what $e^pi$ is? No combination of repeated multiplication, inversion, or taking roots of $e$ will produce this number. (Note, this can be mediated by defining $exp(x) = lim_ntoinfty left(1 + fracxnright)^n$).
The indefinite integral of $frac1x$ is such a natural question that it warrants the invention of a function to fill the gap.
The log laws (and hence exponential laws) turn into lovely applications of various integral rules.
The calculus properties of $ln$ and $exp$ follow immediately from this definition too.
That's why I would teach the integral definition.
answered Jul 29 at 21:00
Theo Bendit
11.8k1841
I'm no expert on maths history, but logarithms are old enough not to have a "historical definition" that meets our standards of what a definition should be. I think the integral definition of the logarithm is the better one to teach, for a few reasons:
What is exponentiation? Even if you define $e$, that may instantly tell you $e^2$ or $e^-frac13$, but how do you tell what $e^pi$ is? No combination of repeated multiplication, inversion, or taking roots of $e$ will produce this number. (Note, this can be mediated by defining $exp(x) = lim_ntoinfty left(1 + fracxnright)^n$).
The indefinite integral of $frac1x$ is such a natural question that it warrants the invention of a function to fill the gap.
The log laws (and hence exponential laws) turn into lovely applications of various integral rules.
The calculus properties of $ln$ and $exp$ follow immediately from this definition too.
That's why I would teach the integral definition.
answered Jul 29 at 21:00
Theo Bendit
11.8k1841
answered Jul 29 at 21:00
Theo Bendit
11.8k1841
answered Jul 29 at 21:00
Theo Bendit
11.8k1841
answered Jul 29 at 21:00
Theo Bendit
11.8k1841
11.8k1841
add a comment |Â
add a comment |Â
up vote
0
down vote
The fact that they call it a "logarithm" implies the must have had a concept that it is the logarithm of some base. So when the defined they must have been using the concept $ln x = y iff e^y = x$. And I even imagine they would be aware that $frac db^xdx = C_b*b^x$ (for rational values of $x$; irrational values would have been poorly understood) so that would figure there must be a base so that $C_b = 1$ and $frac de^xdx = e^x$.
But although that can be the concept and germination of a definition, it can't actually be a practical definition until after they had some way of finding what $e$ would be. And I imagine to do that they had to recognize that $int frac 1x dx$ is a logrithmic function and the value of it's base would be $lim (1 +frac 1n)^n$.
So I would guess, it went in this order 1) $frac db^xdx C(b)*b^x$ for some function $C(b)=limfrac b^h - 1h$. 2) That therefore $C(b) = int_1^b frac 1t dt$ and that $C(b)$ 3) the $e$ so that $C(b) = 1$ is $lim(1 + frac 1n)^n$ then 4) noting $log_e (x) = C(x)$ is an immediate consequence and then the final definition 4) $ln x := log_e x = int_1^xfrac 1t dt = C(x)$.
But I'm just guessing.
answered Jul 29 at 22:07
fleablood
60.2k22575
Guessing isn't very useful for determining the actual history...
– Hans Lundmark
Jul 30 at 7:40
add a comment |Â
up vote
0
down vote
The fact that they call it a "logarithm" implies the must have had a concept that it is the logarithm of some base. So when the defined they must have been using the concept $ln x = y iff e^y = x$. And I even imagine they would be aware that $frac db^xdx = C_b*b^x$ (for rational values of $x$; irrational values would have been poorly understood) so that would figure there must be a base so that $C_b = 1$ and $frac de^xdx = e^x$.
But although that can be the concept and germination of a definition, it can't actually be a practical definition until after they had some way of finding what $e$ would be. And I imagine to do that they had to recognize that $int frac 1x dx$ is a logrithmic function and the value of it's base would be $lim (1 +frac 1n)^n$.
So I would guess, it went in this order 1) $frac db^xdx C(b)*b^x$ for some function $C(b)=limfrac b^h - 1h$. 2) That therefore $C(b) = int_1^b frac 1t dt$ and that $C(b)$ 3) the $e$ so that $C(b) = 1$ is $lim(1 + frac 1n)^n$ then 4) noting $log_e (x) = C(x)$ is an immediate consequence and then the final definition 4) $ln x := log_e x = int_1^xfrac 1t dt = C(x)$.
But I'm just guessing.
answered Jul 29 at 22:07
fleablood
60.2k22575
Guessing isn't very useful for determining the actual history...
– Hans Lundmark
Jul 30 at 7:40
add a comment |Â
up vote
0
down vote
up vote
0
down vote
The fact that they call it a "logarithm" implies the must have had a concept that it is the logarithm of some base. So when the defined they must have been using the concept $ln x = y iff e^y = x$. And I even imagine they would be aware that $frac db^xdx = C_b*b^x$ (for rational values of $x$; irrational values would have been poorly understood) so that would figure there must be a base so that $C_b = 1$ and $frac de^xdx = e^x$.
But although that can be the concept and germination of a definition, it can't actually be a practical definition until after they had some way of finding what $e$ would be. And I imagine to do that they had to recognize that $int frac 1x dx$ is a logrithmic function and the value of it's base would be $lim (1 +frac 1n)^n$.
So I would guess, it went in this order 1) $frac db^xdx C(b)*b^x$ for some function $C(b)=limfrac b^h - 1h$. 2) That therefore $C(b) = int_1^b frac 1t dt$ and that $C(b)$ 3) the $e$ so that $C(b) = 1$ is $lim(1 + frac 1n)^n$ then 4) noting $log_e (x) = C(x)$ is an immediate consequence and then the final definition 4) $ln x := log_e x = int_1^xfrac 1t dt = C(x)$.
But I'm just guessing.
answered Jul 29 at 22:07
fleablood
60.2k22575
The fact that they call it a "logarithm" implies the must have had a concept that it is the logarithm of some base. So when the defined they must have been using the concept $ln x = y iff e^y = x$. And I even imagine they would be aware that $frac db^xdx = C_b*b^x$ (for rational values of $x$; irrational values would have been poorly understood) so that would figure there must be a base so that $C_b = 1$ and $frac de^xdx = e^x$.
But although that can be the concept and germination of a definition, it can't actually be a practical definition until after they had some way of finding what $e$ would be. And I imagine to do that they had to recognize that $int frac 1x dx$ is a logrithmic function and the value of it's base would be $lim (1 +frac 1n)^n$.
So I would guess, it went in this order 1) $frac db^xdx C(b)*b^x$ for some function $C(b)=limfrac b^h - 1h$. 2) That therefore $C(b) = int_1^b frac 1t dt$ and that $C(b)$ 3) the $e$ so that $C(b) = 1$ is $lim(1 + frac 1n)^n$ then 4) noting $log_e (x) = C(x)$ is an immediate consequence and then the final definition 4) $ln x := log_e x = int_1^xfrac 1t dt = C(x)$.
But I'm just guessing.
answered Jul 29 at 22:07
fleablood
60.2k22575
answered Jul 29 at 22:07
fleablood
60.2k22575
answered Jul 29 at 22:07
fleablood
60.2k22575
answered Jul 29 at 22:07
fleablood
60.2k22575
60.2k22575
Guessing isn't very useful for determining the actual history...
– Hans Lundmark
Jul 30 at 7:40
add a comment |Â
Guessing isn't very useful for determining the actual history...
– Hans Lundmark
Jul 30 at 7:40
Guessing isn't very useful for determining the actual history...
– Hans Lundmark
Jul 30 at 7:40
Guessing isn't very useful for determining the actual history...
– Hans Lundmark
Jul 30 at 7:40
add a comment |Â
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9
en.wikipedia.org/wiki/History_of_logarithms
– vadim123
Jul 29 at 20:45
2
Neither is the "real" definition of the natural logarithm. Or both are. They are equivalent---you can start with either and get to the other (and there are other definitions, too, e.g. in terms of a power series). Historically, it is likely that neither is the first definition to appear, either (Napier's tables are probably where logarithms get their modern start). Pedagogically, pick the one that works best for your students, and which best matches with the background they have.
– Xander Henderson
Jul 29 at 20:50
1
@XanderHenderson yeah i know , but my question is which of them is the first historical definition
– El Mouden
Jul 29 at 20:53
2
Please edit your question rather than just clarifying it in a comment.
– Rob Arthan
Jul 29 at 20:58
2
See hsm.stackexchange.com/questions/42/…
– quid♦
Jul 29 at 21:00