5.5 Properties and Laws of Logarithms Do Now: Solve for x. x = 3 x = 1/3 x = 6 x = 12

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  10x = 678 If 10x = 678, what should x be in order to produce 678? x = log(678) because 10log(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. ln (54) = x  ex = 54 If ex = 54, what should x be in order to produce 54? x = ln(54) because eln(54) = 54

Basic Properties of Logarithms Common Logarithms Natural 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 = 1 2. ln 1 = 0 and ln e = 1 3. log 10k = k for every real number k. 3. ln ek = k for every real number k. 4. 10logv=v for every v > 0. 4. elnv=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 aman=am+n Because logarithms represent exponents, it is helpful to review laws of exponents before exploring laws of logarithms. When multiplying like bases, add the exponents. aman=am+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

Using Product and Quotient Laws Given that log 3 = 0.4771 and log 4 = 0.6021, find log 12. Given that log 40 = 1.6021 and log 8 = 0.9031, find log 5. log 12 = log (3•4) = log 3 + log 4 = 1.0792 log 5 = log (40 / 8) = log 40 – log 8 = 0.6990

Power Law of Logarithms For all k and v > 0, log vk = k log v ln vk = k ln v For example… log 9 = log 32 = 2 log 3

Using the Power Law Given that log 25 = 1.3979, find log . Given that ln 22 = 3.0910, find ln 22. log (25¼) = ¼ log 25 = 0.3495 ln (22½) = ½ ln 22 = 1.5455

Simplifying Expressions Logarithmic expressions can be simplified using logarithmic properties and laws. Example 1: Write ln(3x) + 4ln(x) – ln(3xy) as a single logarithm. ln(3x) + 4ln(x) – ln(3xy) = ln(3x) + ln(x4) – ln(3xy) = ln(3x•x4) – ln(3xy) = ln(3x5) – ln(3xy) =

Simplifying Expressions Simplify each expression. log 8x + 3 log x – log 2x2 log 4x2 ¼