7.4 Laws of Logarithms

After we've learned the basic properties of logarithms, here are some laws that all log functions abide by!

Product Law of Logs

log_a(m*n) = log_am + log_an

if a>0, so m>0,
for example:

let m = a^x
log_am = x

and if n>0, then a≠1
let n = a^y

log_an = y

that would mean...

m*n = a^x * a^y
m*n = a^{x + y}

log_a(m*n) = x+y

therefore:
the new proven product law!

log_a(m*n) = log_am + log_an

For Example:
log₆4+ log₆54

if the law applied then it would mean...
log₆4+ log₆54 = log₆(4*54)
log₆4+ log₆54 = log₆(216)
6^x = 216
6^x = 6³
x = 3

therefore:
log₆4+ log₆54 = log₆(216) = 3

Quotient Law of Logs:

log_a(m/n) = log_am - log_an

so if a>0, then m>0
let m = a^x

log_am = x

if n>0, a≠1

let n = a^y

log_an = y

now...

m / n = a^x / a^y
m / n = a^x-y

log_a (m / n) = x - y

therefore our new proven quotient law is:

log_a(m / n) = log_am - log_an

For example:
log₅375-log₅15

if the law applied, that would mean:
log₅375-log₅15 = log₅(375/15)
log₅375-log₅15 = log₅(25)
5^x = 25
5^x = 5²

x = 2

that would mean
log₅375-log₅15 = log₅(25) = 2