Laws of Logarithms

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Laws of Logs:

Laws of Logs This section involves 2 general procedures. Expanding logarithms. Making a small number of terms into more terms so that you can simplify. Collapsing logarithms. Combining many terms into fewer so that you can simplify.

Notes of the Laws of Logs:

Notes of the Laws of Logs Change of Base Formula Product Law: Quotient Law: Power Law:

Change of Base:

Change of Base By far, one of the most useful formulas for logs. This allows us to change any base log into any other base. We usually want to change it into base 10 or e, since our calculators can handle this. Simply divide using the new base, the log of the term by the log of the base. Example…

Change of Base Example:

Change of Base Example Calculate the value of log 5 (45) Graph f(x)=log 2 (3x) Plug in

Why does this work?:

Why does this work? Here is a short proof of why the change of base formula works. I will not ask you to recreate this.

Product Law:

Product Law Log b (uv) = log b (u) + log b (v) This law allows us to change a multiplication problem into addition, or vice versa. When a problem asks you to expand go  When a problem asks you to collapse go 

Product Law Examples:

Product Law Examples Expand and simplify: log 4 (24) Log 4 (4*6)  Log 4 (4) + log 4 (6)  1 + log 4 (2*3)  1 + log 4 (2) + log 4 (3)  1 + ½ + log 4 (3)  1.5+log 4 (3) Done.

Another Product Example:

Another Product Example Collapse and simplify: log 6 (4)+log 6 (12)+log 6 (4.5) log 6 (4*12) + log 6 (4.5)  log 6 (48) + log 6 (4.5)  log 6 (48*4.5)  log 6 (216)  ? 6 3 = 216 therefore log 6 (216) = 3 Done. This does not always happen. We could have ended up with just a log 6 ( ) as an answer.

Quotient Rule:

Quotient Rule Expand: QR:  log 3 (18) – log 3 (6) PR:  log 3 (9*2) – log 3 (3*2) = log 3 (9)+log 3 (2) – [log 3 (3) + log 3 (2)] 2 + log 3 (2) – 1 – log 3 (2) = 1

Another Example:

Another Example Condense: Always go left to right! Log 4 (40) – log 4 (3)  Log 4 (40/3)

Power law:

Power law Expand: log 3 (81) Log 3 (3 4 ) 4Log 3 (3) 4

All Together Now:

All Together Now Product Rule: log 4 (5) + log 4 (x 3 ) + log 4 (y) Power Rule: log 4 (5) + 3log 4 (x) + log 4 (y) Done.

Again:

Again Quotient Rule: Power Rule: Product Rule: