Core Mathematics 2

Core Mathematics 2

1. Exponential and logarithms

   • You need to be familiar with the function y = a^x (a>0) and to know the shape of its graph
     • As an example, look at a table of values for y = 2^x
       • (-3, 1/8), (-2, 1/4), (-1, 1/2), (0, 1), (1, 2), (2, 4), (3, 8)
         • Note that 2^0 = 1 (in fact a^0 = 1 always if a>0) and 2^-3 = 1/(2^3) = 1/8 (a negative index implies the 'reciprocal' of a postive index)
           • Look at the textbook for the graph, it will go through (0,1)
     • Example 1
       • Look at the textbook for the answers
       • a) On the same axes sketch the graph of y = 3^x, y = 2^x and y = 1.5^x
       • b) On another set of axes sketch the graph of y = (1/2)^x and y = 2^x
   • You need to know how to write an expression as a logarithm
     • Log a (n) = x means that a^x = n, where a is called the base of the logarithm
     • Example 2
       • Write as a logarithm 2^5 = 32
         • 2^5 = 32
           • So log 2 (32) = 5
     • Example 3
       • Rewrite as a logarithm
       • a) 10^3 = 1000
         • log 10 (1000) = 3
       • b) 5^4 = 625
         • log 5 (625) = 4
       • c) 2^10 = 1024
         • log 2 (1024) = 10
     • log a (1) = 0 (a > 0)
     • log a (a) = 1 (a > 0)
     • Example 4
       • Find the value of
       • a) log 3 (81)
         • log 3 (81) = 4
       • b) log 4 (0.25)
         • log 4 (0.25) = -1
       • c) log 0.5 (4)
         • log 0.5 (4) = -2
       • d) log a (a^5)
         • log a (a^5) = 5
   • You need to be able to calculate logarithms to the base 10 using your calculator
     • Example 5
       • Find the value of x for which 10^x = 500
         • 10^x = 500
           • So log 10 (500) = x
             • Therefore x = 2.70 (to 3 s.f.)
   • You need to know the laws of logarithms
     • The laws
       • The multiplication law
         • log a (xy) = log a (x) + log a (y)
       • The division law
         • log a (x/y) = log a (x) - log a (y)
       • The power law
         • log a (x)^k = k(log a (x))
     • Example 6
       • Write as a logarithm:
       • a) log 3 (6) + log 3 (7)
         • = log 3 (6*7)
           • = log 3 (42)
       • b) log 2 (15) - log 2 (3)
         • = log 2 (15/3)
           • = log 2 (5)
       • c) 2log 5 (3) + 3log 5 (2)
         • = log 5 (3^2) + log 5 (2^3)
           • = log 5 (9) + log 5 (8)
             • = log 5 (72)
       • d) log 10 (3) - 4log 10 (1/2)
         • = log 10 (3) - log 10 ((1/2)^4)
           • = log 10 (3) - log 10 (1/16)
             • = log 10 (48)
     • Example 7
       • Write in terms of log a (y) and log a (z)
       • a) log a ((x^2) (y) (z)^3)
         • = log a (x^2) + log a (y) + log a (z^3)
           • + 2log a (x) + log a (y) + 3log a (z)
       • b) log a (x/{y^3})
         • = log 3 (x) - log a (y^3)
           • = log 3 (x) - 3log a (y)
       • c) log a ({x¬{y}}/z)
         • ¬ means square root and any thing in {} is square rooted
         • = log a (x¬{y}) - log a (z)
           • = log a (x) + log a (¬{y}) - log a (z)
             • = log a (x) + (1/2)log a (y)
       • d) log a (x/a^4)
         • = log a (x) - log a (a^4)
           • = log a (x) - 4log a (a)
             • = log a (x) - 4
   • You need to be able to solve a^x = y
     • Example 8
       • Solve the equation 3^x = 20, giving your answer to 3 significant figures
         • 3^x = 20
           • log (3^x) = log 20
             • xlog (3) = log (20)
               • x = {log (20)} / {log (3)}
                 • x = 2.73
     • Example 9
       • Solve the equation 7^(x+1) = 3^(x+2)
         • 7^(x+1) = 3^(x+2)
           • (x+1)log (7) = (x+2)log (3)
             • xlog (7) + 1log (7) = xlog (3) + 2log (3)
               • xlog (7) - xlog (3) = 2log (3) - log (7)
                 • x(log (7) - log (3)) = 2log (3) - log (7)
                   • x = {2log (7) - log (7)} / {log (7) - log (3)}
     • Example 10
       • Solve the equation 5^2x + 7(5^2) - 30 = 0
         • Let y = 5^x
           • y^2 + 7y - 30 = 0
             • (y+10)(y-3) = 0
               • y = -10 or y = 3
                 • 5^x = 3
                   • Can't have a negative solution
                   • log (5^x) = log (3)
                     • xlog (5) = log (3)
                       • x = log (3) / log (5)
                         • x = 0.68 (2d.p.)
   • To evaluate a logarithm using your calculator, you sometimes need to change the base of the logarithm
     • Working in base a, suppose that log a (x) = m
       • Writing this as a power: a^m = x
         • Taking logs to a different base b: log b (a^m) = log b (x)
           • Using the power law: mlog b (a) = log b (x)
             • Can be written as: log b (x) = log a (x) * log b (a)
               • This can be written as: log a (x) = {log b (x)} / {log b (a)}
     • log a (b) = 1 / {log b (a)}
     • Therefore log a (b) = {log b (b)} / {log b (a)
       • So log a (b) = 1 / {log b (a)}
     • Example 11
       • Find, to 3 significant figures, the value of log 8 (11)
         • log 8 (11) = {log 10 (11) / log 10 (8) = 1.15
     • Example 12
       • Solve the equation log 5 (x) + 6log x (5) = 5
         • log 5 (x) + 6 / {log 5 (x)} = 5
           • let log 5 (x) = y
             • y + 6 / y = 5
               • y^2 + 6 = 5y
                 • y^2 - 5y + 6 = 0
                   • (y-2)(y-3) = 0
                     • y = 2 or y = 3
                       • log 5 (x) = 3
                         • 5^3 = x
                           • x = 25
                       • log 5 (x) = 2
                         • 5^2 = x
                           • x = 125