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7.4A Logarithms Algebra II Evaluating Log Expressions • We know 22 = 4 and 23 = 8 • But for what value of y does 2y = 6? • Because 22<6<23 you would expect the answer to be between 2 & 3. • To answer this question exactly, mathematicians defined logarithms. Definition of Logarithm to base a • Let a & x be positive numbers & a ≠ 1. • The logarithm of x with base a is denoted by logax and is defined: • logax = y iff y a =x • This expression is read “log base a of x” • The function f(x) = logax is the logarithmic function with base a. • The definition tells you that the equations logax = y and ay = x are equivilant. • Rewriting forms: • To evaluate log3 9 = x ask yourself… • “Self… 3 to what power is 9?” • 32 = 9 so…… log39 = 2 Log form • log216 = 4 • log1010 = 1 • log31 = 0 • log10 .1 = -1 • log2 6 ≈ 2.585 Exp. form • 24 = 16 1 • 10 = 10 0 •3 = 1 • 10-1 = .1 • 22.585 = 6 Evaluate without a calculator • log381 = 4 • Log5125 = 3 • Log4256 = 4 • Log2(1/32) = -5 x •3 = 81 x • 5 = 125 • 4x = 256 x • 2 = (1/32) Evaluating logarithms now you try some! • Log 4 16 = 2 • Log 5 1 = 0 • Log 4 2 = ½ (because 41/2 = 2) • Log 3 (-1) = undefined • (Think of the graph of y=3x) You should learn the following general forms!!! 0 a • Log a 1 = 0 because = 1 1 • Log a a = 1 because a = a • Log a ax = x because ax = ax Natural logarithms •log e x = ln x • ln means log base e Common logarithms •log 10 x = log x • Understood base 10 if nothing is there. Common logs and natural logs with a calculator log10 button ln button • 7.4B Finding Inverses & Graphs • g(x) = log b x is the inverse of • f(x) = bx • f(g(x)) = x and g(f(x)) = x • Exponential and log functions are inverses and “undo” each other • So: g(f(x)) = logb • x b =x log x f(g(x)) = b b = x • 10log2 = 2 x • Log39 = Log3(32)x =Log332x=2x logx • 10 = x x • Log5125 = 3x Ex. 1)Finding Inverses • Find the inverse of: • y = log3x ( write it in exponential form and switch the x & y! ) 3y = x 3x = y Ex. 2) Finding Inverses cont. • Find the inverse of : • Y = ln (x +1) • X = ln (y + 1) • ex = y + 1 • ex – 1 = y Switch the x & y Write in exp form solve for y Graphs of logs • y = logb(x-h)+k • Has vertical asymptote x=h • The domain is x>h, the range is all reals • If b>1, the graph moves up to the right • If 0<b<1, the graph moves down to the right Ex. 3) Graph y =log5(x+2) • y = logb(x-h)+k • Plot easy points (-1,0) & (3,1) • Label the asymptote x=-2 • Connect the dots using the asymptote. X=-2 Ex. 4) y log 4 ( x 2) 1 y log 4 ( x 2) 1 • Ex. 5) Graph y = log1/3x-1 (without parentheses) • Plot (1/3,0) & (3,-2) • Vert line x=0 is asy. • Connect the dots X=0 Assignment