Logarithm
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- 2. 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
- 3. 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.
- 4. 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
- 5. 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
- 6. 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
- 7. 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.
- 8. 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
- 9. 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
- 10. 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.
- 11. 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
- 12. (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.
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