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Transcript of 7.4 Logarithms
7.4 Logarithmsp. 499What you should learn:Goal1Goal2Evaluate logarithmsGraph logarithmic functions7.4 Evaluate Logarithms and Graph Logarithmic FunctionsA3.2.2
Evaluating Log ExpressionsWe 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 aLet a & x be positive numbers & a 1.The logarithm of x with base a is denoted by logax and is defined:logax = y iff ay = xThis expression is read log base a of xThe 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 yourselfSelf 3 to what power is 9?32 = 9 so log39 = 2
Log form Exp. formlog216 = 4log1010 = 1log31 = 0log10 .1 = -1log2 6 2.58524 = 16101 = 1030 = 110-1 = .122.585 = 6
Evaluate without a calculatorlog381 = Log5125 =Log4256 =Log2(1/32) =3x = 815x = 1254x = 2562x = (1/32)434-5
Evaluating logarithms now you try some!Log 4 16 = Log 5 1 =Log 4 2 =Log 3 (-1) =(Think of the graph of y=3x) 20 (because 41/2 = 2) undefined
You should learn the following general forms!!!Log a 1 = 0 because a0 = 1Log a a = 1 because a1 = aLog a ax = x because ax = ax
Natural logarithmslog e x = ln x
ln means log base e
Common logarithmslog 10 x = log x
Understood base 10 if nothing is there.
Common logs and natural logs with a calculatorlog10 button
g(x) = log b x is the inverse off(x) = bx
f(g(x)) = x and g(f(x)) = xExponential and log functions are inverses and undo each other
So: g(f(x)) = logbbx = x f(g(x)) = blogbx = x
10log2 = Log39x =10logx =Log5125x = 2Log3(32)x =Log332x=2xx3x
Finding InversesFind the inverse of:y = log3xBy definition of logarithm, the inverse is y=3x OR write it in exponential form and switch the x & y! 3y = x 3x = y
Finding Inverses cont.Find the inverse of :Y = ln (x +1)X = ln (y + 1) Switch the x & yex = y + 1 Write in exp formex 1 = y solve for y
- Graphs of logsy = logb(x-h)+k Has vertical asymptote x=hThe domain is x>h, the range is all realsIf b>1, the graph moves up to the rightIf 0
Graph y =log5(x+2)Plot easy points (-1,0) & (3,1)Label the asymptote x=-2Connect the dots using the asymptote.X=-2