Product Law of Logs

log

_{a}(m*n) = log

_{a}m + log

_{a}n

**if a>0, so m>0,**

for example:

**let m = a**

^{x}**log**

_{a}

**m = x**

**and if n>0, then a≠1**

**let n = a**

^{y}**log**

_{a}

**n = y**

**that would mean...**

**m*n = a**

^{x}

*** a**

^{y}

m*n = a

m*n = a

^{x + y}

**log**

_{a}

**(m*n) = x+y**

therefore:

the new proven product law!

therefore:

the new proven product law!

__log__

_{a}(m*n) = log_{a}m + log_{a}n**For Example:**

**log**

_{6}**4**

**+ log**

_{6}**54**

if the law applied then it would mean...

if the law applied then it would mean...

**log**

_{6}**4**

**+ log**

_{6}**54 =**

**log**

_{6}**(4*54)**

**log**

_{6}**4**

**+ log**

_{6}**54**

**=**

**log**

_{6}**(216)**

6

^{x}= 216

6

^{x}= 6

^{3}

x = 3

therefore:

**log**

_{6}**4**

**+ log**

_{6}**54**

**=**

**log**

_{6}**(216) = 3**

Quotient Law of Logs:

log

_{a}(m/n) = log

_{a}m - log

_{a}n

let m = a

**so if a>0, then m>0**

let m = a

^{x}

**log**_{a}**m = x**

**if n>0, a≠1**

**let n = a**

^{y}

**log**_{a}**n = y**

now...

now...

**m / n = a**

^{x}

**/ a**

^{y}

m / n = a

m / n = a

^{x-y}

**log**

_{a}

**(m / n) = x - y**

therefore our new proven quotient law is:

therefore our new proven quotient law is:

__log__

_{a}(m / n) = log_{a}m - log_{a}n

For example:

log

For example:

log

_{5}

**375****-log**

if the law applied, that would mean:

_{5}15if the law applied, that would mean:

**log**_{5}**375****-log**_{5}15 =

**log**_{5}(375/15)

loglog

_{5}**375****-log**_{5}15 =**log**

_{5}(25)

**5**^{x }

**= 25**

**5**^{x }**=**

**5**^{2}

^{}

**x = 2**

that would mean

that would mean

**log**_{5}**375****-log**_{5}15 =**log**_{5}(25) = 2

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