# 4.3 Laws of Logarithms

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17-Jan-2016Category

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4.3 Laws of Logarithms

*Laws of Logarithms Just like the rules for exponents there are corresponding rules for logs that allow you to rewrite the log of a product, the log of a quotient, or the log of a power.

*Log of a ProductLogs are just exponentsThe log of a product is the sum of the logs of the factors:logb xy = logb x + logb yEx: log (25 125) = log 25 + log 125

*Log of a QuotientLogs are exponents The log of a quotient is the difference of the logs of the factors: logb = logb x logb y Ex: ln ( ) = ln 125 ln 25

*Log of a PowerLogs are exponents The log of a power is the product of the exponent and the log: logb xn = nlogb x Ex: log 32 = 2 log 3

*Rules for Logarithms These same laws can be used to turn an expression into a single log: logb x + logb y = logb xy logb x logb y = logb nlogb x = logb xn

*logb(xy) = logb x + logb yExpressas a sum and difference of logarithms:= log3A + log3BExampleslogb( ) = logb x logb ylogb xn = n logb x_______________________________Solve: x = log330 log310 = log33Evaluate: = 2 = log3Cx = 1= log3 AB

*Sample Problem Express as a single logarithm:3log7x + log7(x+1) - 2log7(x+2) 3log7x = log7x3 2log7(x+2) = log7(x+2)2log7x3 + log7(x+1) - log7(x+2)2 log7(x3(x+1)) - log7(x+2)2

log7(x3(x+1)) - log7(x+2)2 =

To use a calculator to evaluate logarithms with other bases, you can change the base to 10 or e by using either of the following:For all positive numbers a, b, and x, where a 1 and b 1:Example: Evaluate log4 22

2.2295Change of Base Formula

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