Expanding and Condensing Logarithms

Product Property

= log a M + log a N Ex. Express as a sum of logarithms. 1) log a MN = log b A + log b T 2) log b AT = log M + log A + log T + log H 3) log MATH

= log 5 (193) Ex. Express as a single logarithm 4) log log 5 3 5) log C + log A + log B + log I + log N = log CABIN

Ex. Express as a sum of logarithms, then simplify. 6) log 2 (416) = log log 2 16 = 6 = 2 + 4

Ex. 7 Use log 5 3 = and log 5 7 = to approximate… log 5 (21) = log log 5 7 = = = log 5 (3 7)

Quotient Property

Ex. Express as the difference of logs 8) 9)

Ex. 10 Use log 5 3 = and log 5 7 = to approximate… = log 5 3 – log 5 7 = – =

Power Property

= -5 log b 9 Ex. Express as a product. 11) 12)

Ex. 13 Use log 5 3 = and log 5 7 = to approximate… = log = 2(1.209) = log 5 49 = 2 log 5 7

Ex. 14 Expand log 10 5x 3 y = log log 10 y+ log 10 x 3 = log log 10 y + 3 log 10 x

Ex. 15 Expand Simplify the division. Simplify the multiplication of 4  Change the radical sign to an exponent Express the exponent as a product

Ex. Condense. 16) 17)

Express all products as exponents Simplify the subtraction. Change the fractional exponent to a radical sign. Simplify the addition. Ex 18 Condense

Properties of Logarithms log a 1 = 0 log a a = 1 log a a x = x If log a x= log a ythen x = y because a 0 = 1 because a 1 = a Change-of-Base Product Property Quotient Property Power Property