Presentation on theme: "5.5 Properties and Laws of Logarithms"— Presentation transcript:
15.5 Properties and Laws of Logarithms Do Now: Solve for x.x = 3x = 1/3x = 6x = 12
2Consider 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 = 678If 10x = 678, what should x be in order to produce 678?x = log(678) because 10log(678) = 678
3And 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 = 54If ex = 54, what should x be in order to produce 54?x = ln(54) because eln(54) = 54
4Basic Properties of Logarithms Common LogarithmsNatural Logarithms1. 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 = 13. 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! **
5Example 1: Solving Equations Using Properties Use the basic properties of logarithms to solve each equation.
6Laws 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+nWhen dividing like bases, subtract the exponents.
7Product and Quotient Laws of Logarithms For all v,w>0,log(vw) = log v + log wln(vw) = ln v + ln w
8Using Product and Quotient Laws Given that log 3 = and log 4 = , find log 12.Given that log 40 = and log 8 = , find log 5.log 12 = log (3•4) = log 3 + log 4 =log 5 = log (40 / 8) = log 40 – log 8 =
9Power Law of Logarithms For all k and v > 0,log vk = k log vln vk = k ln vFor example…log 9 = log 32 = 2 log 3
10Using the Power Law Given that log 25 = 1.3979, find log . Given that ln 22 = , find ln 22.log (25¼) = ¼ log 25 =ln (22½) = ½ ln 22 =
11Simplifying 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)=
12Simplifying Expressions Simplify each expression.log 8x + 3 log x – log 2x2log 4x2