9 5 Properties of Logarithms Laws of Logarithms

9. 5 Properties of Logarithms

Laws of Logarithms ª 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. 2

Log of a Product ª Logs are just exponents ª The log of a product is the sum of the logs of the factors: ©logb xy = logb x + logb y ©Ex: log (25 · 125) = log 25 + log 125 3

Log of a Quotient ª Logs are exponents ª The log of a quotient is the difference of the logs of the factors: © logb © Ex: ln ( = logb x – logb y ) = ln 125 – ln 25 4

Log of a Power ª Logs are exponents ª The log of a power is the product of the exponent and the log: © logb xn = n∙logb x © Ex: log 32 = 2 ∙ log 3 5

Rules for Logarithms ª 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 © n∙logb x = logb xn 6

Examples logb(xy) = logb x + logb y logb( ) = logb x – logb y logb xn = n logb x ________________ as a sum and difference of logarithms: Express = log=3 Alog + 3 log AB 3 B – log 3 C Solve: x = log 330 – log 310 Evaluate: = log 33 x=1 = 2 = 7

Sample Problem ª Express as a single logarithm: 3 log 7 x + log 7(x+1) - 2 log 7(x+2) § 3 log 7 x = log 7 x 3 § 2 log 7(x+2) = log 7(x+2)2 log 7 x 3 + log 7(x+1) - log 7(x+2)2 log 7(x 3·(x+1)) - log 7(x+2)2 § log 7(x 3·(x+1)) - log 7(x+2)2 = 8

Change of Base Formula For all positive numbers a, b, and x, where a ≠ 1 and b ≠ 1: 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: Example: Evaluate log 4 22 ≈ 2. 2295