logn(x)/logn(y) = logy(x) ~ Is this a thing?

[anonybystander]

I am not a mathematician by any standard, but have come across something which I find pretty cool and have no idea if this is something that exists already or perhaps something new(ish).

As the title states the formula is logn(x) / logn(y) = logy(x).

An example would be log2(e)/log2(Pi) = logPi(e).

This results in Pi^0.873568526 ~ e^1.4472988585000301

For fun I did log2(sqrt(163))/log2(Pi) = sqrt(163)^0.4494644773384984909610879 ~ Pi^2.224869929479980564572834368611955176788795867307990637817

Again I am no maths savvy so getting the opinions of some actual mathematicians would be helpful. Thanks in advance for such a dumb question.

 

[tkhunny]

I am not a mathematician by any standard, but have come across something which I find pretty cool and have no idea if this is something that exists already or perhaps something new(ish).

As the title states the formula is logn(x) / logn(y) = logy(x).

An example would be log2(e)/log2(Pi) = logPi(e).

This results in Pi^0.873568526 ~ e^1.4472988585000301

For fun I did log2(sqrt(163))/log2(Pi) = sqrt(163)^0.4494644773384984909610879 ~ Pi^2.224869929479980564572834368611955176788795867307990637817

Again I am no maths savvy so getting the opinions of some actual mathematicians would be helpful. Thanks in advance for such a dumb question.

I'm glad you find it fun. Keep exploring!

 

[Dr.Peterson]

I am not a mathematician by any standard, but have come across something which I find pretty cool and have no idea if this is something that exists already or perhaps something new(ish).

As the title states the formula is logn(x) / logn(y) = logy(x).

An example would be log2(e)/log2(Pi) = logPi(e).

This results in Pi^0.873568526 ~ e^1.4472988585000301

For fun I did log2(sqrt(163))/log2(Pi) = sqrt(163)^0.4494644773384984909610879 ~ Pi^2.224869929479980564572834368611955176788795867307990637817

Again I am no maths savvy so getting the opinions of some actual mathematicians would be helpful. Thanks in advance for such a dumb question.

Yes, log_n(x)/log_n(y) = log_y(x) is "a thing"; it's called the change of base formula for logarithms. You can read about it here, for example: https://www.purplemath.com/modules/logrules5.htm, or https://proofwiki.org/wiki/Change_of_Base_of_Logarithm . The latter site gives a proof.

You have slightly misstated your result; I think you meant that Pi^0.873568526 ~ e. I'm not sure what you are saying e^1.4472988585000301 is.

 

[anonybystander]

Yes, log_n(x)/log_n(y) = log_y(x) is "a thing"; it's called the change of base formula for logarithms. You can read about it here, for example: https://www.purplemath.com/modules/logrules5.htm, or https://proofwiki.org/wiki/Change_of_Base_of_Logarithm . The latter site gives a proof.

You have slightly misstated your result; I think you meant that Pi^0.873568526 ~ e. I'm not sure what you are saying e^1.4472988585000301 is.

I see thank you heaps!! I was meaning that e^1.4472988585000301 = Pi (sorry forgot to put the answer) Thank you again for helping me understand more!

 

[mmm4444bot]

I was meaning that e^1.4472988585000301 = Pi (sorry forgot to put the answer)

That e͏quation is not true, either. :-(

 

[JeffM]

I am not a mathematician by any standard, but have come across something which I find pretty cool and have no idea if this is something that exists already or perhaps something new(ish).

As the title states the formula is logn(x) / logn(y) = logy(x).

An example would be log2(e)/log2(Pi) = logPi(e).

This results in Pi^0.873568526 ~ e^1.4472988585000301

For fun I did log2(sqrt(163))/log2(Pi) = sqrt(163)^0.4494644773384984909610879 ~ Pi^2.224869929479980564572834368611955176788795867307990637817

Again I am no maths savvy so getting the opinions of some actual mathematicians would be helpful. Thanks in advance for such a dumb question.

This is quite parallel with Dr P's comment, but you need to be much more careful with notation.

You sometimes get answers in logs to a non-standard base. (The standard bases are e and 10.) Calculators (or tables) will give you approximate values for logs to standard bases, but are no help with non-standard bases. Here comes the utility of the change of base formula.

\(\displaystyle x = log_a(b)\) where a is not a standard base. Let d be a standard base.

\(\displaystyle x = log_a(b) \implies a^x = b \implies log_d(a^x) = log_d(b) \implies \\

x log_d(a) = log_d(b) \implies x = \dfrac{log_d(b)}{log_d(a)}. \\

\text {But, by hypothesis, } x = log_a(b). \\

\therefore log_a(b) = \dfrac{log_d(b)}{log_d(a)}.\)

 

[anonybystander]

That e͏quation is not true, either. :-(

Sorry forgot a 1 ~ e^1.14472988585000301 = Pi

 

[anonybystander]

This is quite parallel with Dr P's comment, but you need to be much more careful with notation.

You sometimes get answers in logs to a non-standard base. (The standard bases are e and 10.) Calculators (or tables) will give you approximate values for logs to standard bases, but are no help with non-standard bases. Here comes the utility of the change of base formula.

\(\displaystyle x = log_a(b)\) where a is not a standard base. Let d be a standard base.

\(\displaystyle x = log_a(b) \implies a^x = b \implies log_d(a^x) = log_d(b) \implies \\

x log_d(a) = log_d(b) \implies x = \dfrac{log_d(b)}{log_d(a)}. \\

\text {But, by hypothesis, } x = log_a(b). \\

\therefore log_a(b) = \dfrac{log_d(b)}{log_d(a)}.\)

Thank you for the explanation ~ math seems to blow my mind more and more as I get older haha. I wish they found a way to teach it better in educational systems.