*Exponent Laws*we have learned in previous years,

*translate*into the Laws of Logarithms.

1. Product Law of Logarithms

__l____og___{a}(m*n) = log_{a}m + log_{a}n**a>0, m>0, n>0, a≠1**

__Proof:__**let m = a**

let n = a

^{x}let n = a

^{y}

**Now...turn these into logarithms:**

**log**

_{a}

**m = x**

log

log

_{a}

**n = y**

**So...**

**m*n = a**

^{x}

*** a**

^{y}

m*n = a

m*n = a

^{x+y}

**log**

_{a}

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

__log___{a}(m*n) = log_{a}m + log_{a}n

**Example:**

**Evaluate... log**

log

log

_{8}4 + log_{8}16log

_{8}(4*16)log

_{8}64**8**

^{x}=64**8**

^{x}=8^{2}**x=2**

**2. Quotient Law of Logarithms**

__l____og___{a}(m/n) = log_{a}m - log_{a}n**a>0, m>0, n>0, a≠1**

__Proof:__**let m = a**

let n = a

^{x}let n = a

^{y}

**Now...turn these into logarithms:**

**log**

_{a}

**m = x**

log

log

_{a}

**n = y**

**So...**

**m/n = a**

^{x}

**/ a**

^{y}

m/n = a

m/n = a

^{x-y}

**log**

_{a}

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

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

**Example:**

**Evaluate... log**

_{3}405-log_{3}5**log**

_{3}(405/5)**log**

_{3}81**3**

^{x}=81**3**

^{x}=3^{4}**x=4**

**3. Power Law of Logarithms**

__l____og___{a}(n^{m}) = mlog_{a}n

**Proof:**

**let n=a**

^{x}

**therefore:**

**log**

_{a}n=x

**n**

^{m}= (a^{x})^{m}**n**

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

_{a}(n^{m})=x*m**log**

_{a}(n^{m})=mlog_{a}n

**Example:**

**log**

=4log

_{3}9^{4}=4log

_{3}9**=4log**

_{3}3^{2}**=4(2)log**

_{3}3**=4(2)(1)**

**=8**

**Today...we also learned about the Change of Base Formula:**

**it helps us solve problems like log**_{2}10

**WAIT A SECOND...There's still a way to solve it...**

**We shall use the Change of Base Formula!**

**log**

_{b}X=log_{a}X/log_{a}b

**log**

_{2}10=log10/log2 (assume common logarithms...a=10)**now use your calculator:**

**log**

_{2}10=3.32 (approximate)**2**

^{3.32}=10

**If you still don't understand...watch this video =)**

^{}

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