Expanding and Condensing Logarithms

Common Logarithm log10 = log A common logarithm is a logarithm that is base 10. When a logarithm is base 10, we don’t write the base. log10 = log We like base 10 because we can evaluate it in our calculator. (Use the LOG button.)

Evaluate with a calculator = 1 21) log10 10 = 0.7959 22) 2 log10 2.5 no solution 23) log10 (-2) Remember this means 10? = -2

More Properties of Logarithms If loga x = loga y then x = y

Product Property

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

Ex. Express as a single logarithm 4) log5 19 + log5 3 = log5 (19•3) 5) log C + log A + log B + log I + log N = log CABIN

Ex. Express as a sum of logarithms, then simplify. = log2 4 + log216 = 2 + 4 = 6

Ex. 7 Use log53 = 0.683 and log57 = 1.209 to approximate… = log5 3 + log5 7 = 0.683 + 1.209 = 1.892

Quotient Property

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

Ex. 10 Use log53 = 0.683 and log57 = 1.209 to approximate… = log5 3 – log5 7 = 0.683 – 1.209 = -0.526

Power Property

Ex. Express as a product. = -5 • logb9 11) 12)

Ex. 13 Use log53 = 0.683 and log57 = 1.209 to approximate… = 2(1.209) = 2.418

Ex. 14 Expand log105x3y = log105 + log10x3 + log10y = log105 + 3 log10x + log10y

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)

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

Properties of Logarithms because a0 = 1 logaa = 1 because a1 = a logaax = x If loga x= loga y then x = y Product Property Quotient Property Power Property Change-of-Base

Warning!! Be careful!!