Presentation on theme: "5.5 Properties and Laws of Logarithms Do Now: Solve for x. x = 3 x = 12 x = 6 x = 1/3."— Presentation transcript:
5.5 Properties and Laws of Logarithms Do Now: Solve for x. x = 3 x = 12 x = 6 x = 1/3
Consider some more examples… Without evaluating log (678), we know the expression means the exponent to which 10 must be raised in order to produce 678. log (678) = x 10 x = 678 If 10 x = 678, what should x be in order to produce 678? x = log(678) because 10 log(678) = 678
And with natural logarithms… Without evaluating ln (54), we know the expression means the exponent to which e must be raised in order to produce 54. If e x = 54, what should x be in order to produce 54? x = ln(54) because e ln(54) = 54 ln (54) = x e x = 54
Basic Properties of Logarithms Common LogarithmsNatural Logarithms 1. log v is defined only when v > 0. 1. ln v is defined only when v > 0. 2. log 1 = 0 and log 10 = 12. ln 1 = 0 and ln e = 1 3. log 10 k = k for every real number k. 3. ln e k = k for every real number k. 4. 10 logv =v for every v > 0.4. e lnv =v for every v > 0. ** NOTE: These properties hold for all bases – not just 10 and e! **
Example 1: Solving Equations Using Properties Use the basic properties of logarithms to solve each equation.
Laws of Logarithms Because logarithms represent exponents, it is helpful to review laws of exponents before exploring laws of logarithms. When multiplying like bases, add the exponents. a m a n =a m+n When dividing like bases, subtract the exponents.
Product and Quotient Laws of Logarithms For all v,w>0, log(vw) = log v + log w ln(vw) = ln v + ln w