Logarithm
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Logarithm

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    Logarithm Logarithm Presentation Transcript

    • T- 1-855-694-8886 Email- info@iTutor.com By iTutor.com
    •  Logarithmic functions are the inverses of exponential functions, and any exponential function can be expressed in logarithmic form.  Similarly, all logarithmic functions can be rewritten in exponential form.  Logarithms are really useful in permitting us to work with very large numbers while manipulating numbers of a much more manageable size.  Every exponential function f(x) = a x, with a > 0 and a ≠ 1. is a one-to-one function, therefore has an inverse function(f-1).  The inverse function is called the Logarithmic function with base a and is denoted by loga Let a be a positive number with a ≠ 1. The logarithmic function with base a, denoted by loga is defined by: Loga x = y a y = Х Clearly, Loga Х is the exponent to which the base a must be raised to give Х
    • y = loga x if and only if x = a y The logarithmic function to the base a, where a > 0 and a 1 is defined: 2 416 exponential form logarithmic form Convert to log form: 216log4 Convert to exponential form: 3 8 1 log2 8 1 2 3 When you convert an exponential to log form, notice that the exponent in the exponential becomes what the log is equal to.
    •  f(x) = 10x is an exponential function where the base is 10 and the exponent is x  Let us write this as: y = 10x Here the power x is the input and the quantity y is the output. The domain is the set of x-values and the range is the set of y- values. A similar statement made by using the quantity y as the input and the power x as the out put is called a logarithmic statement. When you input a quantity y, what will be the power of the base 10 to obtain y? The answer is x  To write this in the proper function form, we exchange x and y.  The statement y = log10x is called a logarithmic function. Log10y = x The logarithm of y to the base 10 is x
    •  Find the value of: 5log 5 5log 5 x is obtained by raising the base tThe qua o the pn otit 55y wer x 5 5x 1 2 5 5x 1 2 x 5log 5 1 2
    • Find the value of: 6 6log 3 6 6is obtained by raising the base 6 to theThe quantit powery log 36x 6log 36 6 x log6x = log636 Since the bases are the same, x = 36 6 3log 6 6 36 2 2 3 3 3 Evaluate: 1 (a)log 8 log 4 ( )log 27 log 3 ( ) log 81 4 b c 2log 8 4 2log 32 5 3 27 log 3 3log 9 2 1 4 3log 81 3log 3 1
    •  Obtain ordered pairs and graph f(x) = log10(x) x 0. 1 0.2 0.4 0.8 1 2 3 4 5 y -1 -0.7 -0.4 -0.1 0 0.3 0.48 0.6 0.7 0.80.60.40.2 0 1.0 1.2 1.4 1.6 1.8 -0.2 -0.4 -0.6 -0.8 0.2 0.4 0.6 0.8 -1.0 2.0 2.2 2.4 2.8 2.8 3.0 (0.1, -1) (0.2, -0.7) (0.4, -0.4) (0.8, -0.1) (1, 0) (2, 0.3) x = 0 is a vertical asymptote for this graph.
    • f(x) = 2x g(x) = logx2 -2 0.25 0.25 -2 -1 0.5 0.5 -1 0 1 1 0 1 2 2 1 2 4 4 2 3 8 8 3 f(x) = 2x g(x) =log2x y = x (1,0) (0,1) x y
    • f(x) = bx Domain: (-∞, ∞) Range: (0, ∞) g(x) = logbx Doman: (0, ∞) Range: (-∞, ∞) f(x) = 2x g(x) =log2x y = x (1,0) (0,1) x y
    •  The graph of g(x) = log2(x – h) + k can be obtained by shifting the graph of f(x) = log2(x) h units horizontally and k units vertically.  Use the graph of f(x) = log2(x) to obtain the graph of g(x) = log2(x – 1) + 2 0-1-2-3-4-5 1 2 3 4 5 -1 -2 -3 -4 1 2 3 4 f(x) g(x) Here h = 1 and k = 2 The graph of f(x) = log2(x) shifts 1 unit to the right and 2 units up x = 1 is a vertical asymptote.
    • Example: A sum of $500 is invested at an interest rate 9%per year. Find the time required for the money to double if the interest is compounded according to the following method. a) Semiannual b) continuous Solution: (a) We use the formula for compound interest with P = $5000, A (t) = $10,000r = 0.09, n = 2, and solve the resulting exponential equation for t. (Divide by 5000) (Take log of each side) (bring down the exponent) (Divide by 2 log 1.045) t ≈ 7.9 The money will double in 7.9 years. (using a calculator) 10000 2 09.0 15000 2t 2045.1 2t 21.04521log 2t 045.1log2)(logt 2log1.045log2t
    • (b) We use the formula for continuously compounded interest with P = $5000, A(t) = $10,000, r = 0.09, and solve the resulting exponential equation for t. 5000e0.09t = 10,000 e 0.091 = 2 (Divide by 5000) In e 0.091 = In 2 (Take 10 of each side) 0.09t = In 2 (Property of In) t=(In 2)/(0.09) (Divide by 0.09) t ≈7.702 (Use a calculator) The money will double in 7.7 years.
    • Call us for more information: www.iTutor.com 1-855-694-8886 Visit