1) Write in exponential form. log 27 9 = x 3) Evaluate. Warm-Up 2) Write in logarithmic form. 5 x = ) Write the Equation that models this situation: The current world population of Oompa-loompas is 7,986. If the Ooompa-loompa population grows steadily at a rate of 19% per year, what will their total population be in 10 years?

U3 Properties of Logarithms

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

Remember Common Logarithm?? log 10 Is the “common” logarithm This is the LOG button on the calculator Sometimes we’re lazy and we don’t even write the base… log 10 7 = log 7

Basic Properties of Logarithms 1) log a 1 = 0 2) log a a = 1 because a 0 = 1 because a 1 = a

3) One-to-one Property: If log a x = log a y then x= y

Product Property = log a M + log a N1) log a MN = log b A + log b T2) log b AT = log M + log A + log T + log H3) log MATH Express as a sum of logarithms.

Express as a single logarithm = log 5 (19*3) Ex. 4) log log 5 3 5) log C + log A + log B + log I + log N = log CABIN

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

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

Quotient Property

Express as the difference of logs Ex. 1) 2)

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

Power Property

Express as a product. = -5 * log b 9 Ex. 4) 5)

Use log 5 3 = and log 5 7 = to approximate… Ex. 6 = log = 2(1.209) = log 5 49 = 2 log 5 7

Ex. 7 log 10 5x 3 y log log 10 y+ log 10 x 3 log log 10 y+ 3 log 10 x Expand.

Expand Simplify the division. Simplify the multiplication of 4  Change the radical sign to an exponent Express the exponent as a product Ex. 8

Ex. Condense. 9) 10)

Condense Express all products as exponents Simplify the subtraction. Change the fractional exponent to a radical sign. Simplify the addition. Ex 11

Change-of-Base this allows us to change from any base to base 10

Ex. 1 Evaluate. Round to four decimal places. Be careful about the parenthesis in the calculator… log(7) / log(3) =

Ex. 2 Evaluate. Round to four decimal places. =