Logarithms: 7 Important Logarithmic Identities

This short video presents seven important logarithmic identities. These can be used in the simplification of algebraic and logarithmic equations.

[patrickpablo]

I use these three:
1) log_b_a = 1 iff a = b
2) log_b_a * log_d_c = log_d_a * log_b_c
3) c*log_b_a = x*log_b^z_a^y for all x*y/z=c

[Apply the usual restrictions to valid inputs.]

You can get everything else from there.

I also find writing "log" all the time confusing, so for "logarithm of a to the base b", instead of writing log_b_a like I did above, I simply write a//b (sort of like how some programming languages use ** for exponentiation). Then the 3 rules above become:

1) a//b = 1 iff a = b
2) a//b * c//d = a//d * c//b
3) c*(a//b) = x*((a^y)//(b^z)) for all x*y/z=c

To me this is easier to remember and to work with since logs have some similarity to fractions. With fractions for example we have 1*1=(a/a)*(b/b)=(a/b)*(b/a) showing that (b/a) is (a/b)^-1. With logarithms we have almost the exact same derivation: 1*1=(a//a)*(b//b)=(a//b)*(b//a), showing that (b//a) is (a//b)^-1. (The same symmetry holds for the change of base formula. The fractions equivalent would be For logarithms, it's Since (c//b)=(b//c)^-1 (from earlier), then (a//b)=(a//c)/(b//c) which is the usual way the change of base formula is given, "the logarithm of a to the base b equals the logarithm of a to the base c, divided by the logarithm of b to the base c."

Hope this helps some other people, too! :)

[og_neso]

Underrated!! Thanks so much!! Loved the vid! Slept off at college math class today but this really really helped! Thanks:)
Wont be falling asleep again!

[adityashelke]

Thank you sir, for your excellent explanation. It made my concepts crystal clear.

[arunkumar_from_mathura]

Hey you its very very helpful for me 🎩
😁
👕👍Great!
👖

It is awesome