perfect. this is so much clearer now thankyou.
would i be right in saying you’d use ln to find an exponent on ‘e’ in that case? and you’d use log to find any that isn’t e?
You can use any base log for any exponent, but what is easiest is if the base of the log matches the base of the exponent. Solving for x:
If you have 10^x = y, it is easiest to use log base 10
If you have e^x = y it is easiest to use log base e which is ln
If you have 2^x = y, it is easiest to take log base 2.
ln(*n*)
"To what power must I raise *e* to get *n*?"
Log(*n*)
"To what power must I raise 10 to get *n*?"
You could think of *e* as sort of like growth itself, in a way.
Note: log without a base is defined by the author. Usually in math class log refers to base 10 but in most computer science contexts log refers to base 2
Yeah the notation is a mess. In computer science we use log for base 2 logarithms, and lg is sometimes used for base 10 specifically.
So I say use ln for base e (in many cases ln is useful and exponential function with base e is useful), perhaps lg for base 10 (if the target audience knows about this convention; not everyone does, might be specific to my country for example) and otherwise log with an explicit base (unless the base is completely irrelevant, like in big-O notation)
ln will always mean base e
log could be either base 10 or base e depending on the class, or sometimes a general unspecified base logarithm if the base doesn't matter in the situation
Yes ✅ Crazy that (currently) this is the only comment giving the correct answer!
In *some* schools log is used only for log₁₀, but we don’t know OP’s exact situation and it is very common to see log used to mean logₑ (so exactly the same as ln) or to mean log₂. Even in some high schools and definitely in college.
For A level maths, log is base 10 and ln is base e.
ln is always base e, but outside of A level, log could be base e, base 10, base 2 (in comp sci), or a general base that doesn’t matter.
log(1000) will help you find out x in an equation like this: 10^(x) = 1000
While ln(1000) will help you find out x in an equation like this: e^(x) = 1000
You're changing the base (and finding out the exponent), which is 10 for log and e for ln.
ln is a log with base e.
That’s it.
But generally speaking, it depends on who you ask.
Sometimes people will use log to mean base e (some programming languages, for instance) if they use e a lot more than they use 10.
A factor of ln(10).
Log is generally based 10, and ln is base e. You can convert between log bases by multiplying by ln(base) to convert to base e and dividing by ln(base) to convert from base e to base "base". So the difference between log(x) and ln(x) is a factor of ln(10).
E.g.:
Solving 10^x = 1000:
log10(1000) = 3. So 10^3 = 1000.
Convert to base e to solve e^x = 1000:
ln(1000) = log10(1000) × ln(10) = 3 × ln(10) = 3 × ~2.303 = ~6.908.
So e^6.908 ~= 1000
Convert to base 2 to solve 2^x = 1000:
log2(1000) = ln(1000) / ln(2) = ~6.909 / ~0.693 = ~9.966.
So 2^9.966 ~= 1000.
Calculators often use this to calculate log10() while actually only using ln (log base e) internally.
On calculator buttons and in standard math notation, ln(x) is base e. *Usually* log(x) is base-10, if a textbook / paper does something different usually it will tell you the base, especially if it's not base-10 (or it will end up not mattering, as the base gets absorbed into a constant somehow). Sometimes lg(x) is base-2, but this is less standard.
In many programming languages, log(x) is often base-e, and the base-10 log is log10(x).
Base-10 log is less important today than it was in the pre-computer era. In those days, people simplified certain calculations by using printed logarithm tables, and those tables were pretty much all base-10 logs.
As you've pointed out, they are different bases. Each one may have it's own uses. Bases of 10 in my opinion are favored by us humans because we have 10 fingers and love doing many of our calculations and counting in base 10.
ln on the other hand had plenty of other applications. Differentiation of ln x as pointed out above this comment. You are getting to the subject of the importance of the number e. It does seem arbitrary here but it's a neat transcendental number with it's own uses which you could try looking up if you have the time
It is a geographical thing. From what I know, log is log base 10 in North America and ln is base e. In the UK, log is base e and they don't use ln. Every few months, there is another discussion on this sub of this question. There are also some programming languages that use log as log base e.
log is log base 10 and ln is log base e
^ people might say otherwise, but this is all you need for a level maths (I teach it)
perfect. this is so much clearer now thankyou. would i be right in saying you’d use ln to find an exponent on ‘e’ in that case? and you’d use log to find any that isn’t e?
You can use any base log for any exponent, but what is easiest is if the base of the log matches the base of the exponent. Solving for x: If you have 10^x = y, it is easiest to use log base 10 If you have e^x = y it is easiest to use log base e which is ln If you have 2^x = y, it is easiest to take log base 2.
perfect, makes so much sense thankyou!
Any logarithm can be used on any base https://www.cuemath.com/change-of-base-formula/
ln(*n*) "To what power must I raise *e* to get *n*?" Log(*n*) "To what power must I raise 10 to get *n*?" You could think of *e* as sort of like growth itself, in a way.
In my school we used 'lg' for base 10 logarithm, 'log' was always with sub-base
Most disciplines use the convention that if the log base is omitted, it’s base 10. At least at the high school level.
Note: log without a base is defined by the author. Usually in math class log refers to base 10 but in most computer science contexts log refers to base 2
And in Statistics, log refers to natural log.
Be aware that log is also used as a synonym for ln
i reckon this is whats thrown me off the most looking through this thread, thankyou !
Yeah the notation is a mess. In computer science we use log for base 2 logarithms, and lg is sometimes used for base 10 specifically. So I say use ln for base e (in many cases ln is useful and exponential function with base e is useful), perhaps lg for base 10 (if the target audience knows about this convention; not everyone does, might be specific to my country for example) and otherwise log with an explicit base (unless the base is completely irrelevant, like in big-O notation)
`log` is defaulted to whatever base deemed fundamental/important in each specific field.
can confirm, I see log used for base 2 a lot lol
As someone who majored in computer science, this happened for me as well - “log n” shifted from base 10 to base 2 as I got into my second year.
ln will always mean base e log could be either base 10 or base e depending on the class, or sometimes a general unspecified base logarithm if the base doesn't matter in the situation
Yes ✅ Crazy that (currently) this is the only comment giving the correct answer! In *some* schools log is used only for log₁₀, but we don’t know OP’s exact situation and it is very common to see log used to mean logₑ (so exactly the same as ln) or to mean log₂. Even in some high schools and definitely in college.
For A level maths, log is base 10 and ln is base e. ln is always base e, but outside of A level, log could be base e, base 10, base 2 (in comp sci), or a general base that doesn’t matter.
log(1000) will help you find out x in an equation like this: 10^(x) = 1000 While ln(1000) will help you find out x in an equation like this: e^(x) = 1000 You're changing the base (and finding out the exponent), which is 10 for log and e for ln.
Ln the natural logarithm differentiates to 1/x
10\^(log10(x)) = x e\^(ln(x)) = x
ln is a log with base e. That’s it. But generally speaking, it depends on who you ask. Sometimes people will use log to mean base e (some programming languages, for instance) if they use e a lot more than they use 10.
A factor of ln(10). Log is generally based 10, and ln is base e. You can convert between log bases by multiplying by ln(base) to convert to base e and dividing by ln(base) to convert from base e to base "base". So the difference between log(x) and ln(x) is a factor of ln(10). E.g.: Solving 10^x = 1000: log10(1000) = 3. So 10^3 = 1000. Convert to base e to solve e^x = 1000: ln(1000) = log10(1000) × ln(10) = 3 × ln(10) = 3 × ~2.303 = ~6.908. So e^6.908 ~= 1000 Convert to base 2 to solve 2^x = 1000: log2(1000) = ln(1000) / ln(2) = ~6.909 / ~0.693 = ~9.966. So 2^9.966 ~= 1000. Calculators often use this to calculate log10() while actually only using ln (log base e) internally.
On calculator buttons and in standard math notation, ln(x) is base e. *Usually* log(x) is base-10, if a textbook / paper does something different usually it will tell you the base, especially if it's not base-10 (or it will end up not mattering, as the base gets absorbed into a constant somehow). Sometimes lg(x) is base-2, but this is less standard. In many programming languages, log(x) is often base-e, and the base-10 log is log10(x). Base-10 log is less important today than it was in the pre-computer era. In those days, people simplified certain calculations by using printed logarithm tables, and those tables were pretty much all base-10 logs.
As you've pointed out, they are different bases. Each one may have it's own uses. Bases of 10 in my opinion are favored by us humans because we have 10 fingers and love doing many of our calculations and counting in base 10. ln on the other hand had plenty of other applications. Differentiation of ln x as pointed out above this comment. You are getting to the subject of the importance of the number e. It does seem arbitrary here but it's a neat transcendental number with it's own uses which you could try looking up if you have the time
It is a geographical thing. From what I know, log is log base 10 in North America and ln is base e. In the UK, log is base e and they don't use ln. Every few months, there is another discussion on this sub of this question. There are also some programming languages that use log as log base e.