Exponentials and Logarithms Section 9.0A   Applications involving population, radioactive decay, carbon-dating, earthquakes and the decibel scale use exponential and logarithm properties. Recall a Function is a correspondence between two sets; the first set are x values, and the second set are y values. For each x value, there is exactly one y value.   Example 1:            y = 2x + 3 When          x = 1,     y = 5         x = 0,     y = 3 x is the independent variable, y the dependent variable. This function is a linear function, the equation is of a line, with constant slope = 2           Slope =    rise     = change in y     = 5 – 3    = 2                         Run        change in x       1 – 0   An exponential function is of the form:                             y = bx   ; b > 0 ; b 1   The base is b, and is constant.  The independent variable x, is the exponent.   Example 2:  Sketch                      y = 3x Exponential functions are models for population, inflation, carbon dating and radioactive decay.  The sharp rise in EX2 graph indicates exponential growth. The base b is any real number, not 1.  It can be rational (can be put into a fraction (ratio), thus it is either a terminating or repeating decimal).  Or irrational, not terminating or repeating decimal.   Examples of irrational numbers: , ,  ~ (pi) . . There is another irrational number that occurs in nature important enough to have a special name,          e = 2.71828182. . . .   Example3:  Use calculator to find         y = ex  natural exponential function. When         x = 1;              y =  e   = 2.718 . .         x = 2               y =  e2 = 7.389 . .         x = -1              y = e-1 = 0.3678 . .         x = -2              y = e-2   = 0.1353         x = 3               y = e3   = 20.085 Logarithms are inverses of the exponential function. What is the exponent of 10 to get 100? to get 1000?  To get 346?   102 = 100       so     x = 2 103 = 1000     so       x = 3 10x = 346                  x = ? We can’t be precise, but we can state          2 < x < 3.   Definition of logarithm:  logb u = v    means            bv = u So we can rewrite          10x = 346      as      log10346 = x Since our number system is base 10, log of base 10 is called the common log and we do not have to write the subscript. 10x = 346   can be rewritten as        log346 = x We can use our calculators to solve.   Example 3:   3 raised to what number gives us 9?                   3v = 9     Or              log39 = v;        v = 2   Find u:             a) u = log28                         b) log5u = -2                         c) logu9 = 2   Sketch          y = log x          which means                  10y = x Logarithms are used in the measurement of earthquakes (the Richter Scale) and sound (decibel scale).  It measures small increments of x with large differences in y for 0 < x < 1 but for x > 1 the y values are compressed. The natural log function has base e, the natural base.         y = loge x     is written          y = ln x   can be rewritten                  x =  = ey   Note: the independent variable in a log function is greater than 0!   Example 4:   Use calculators to Find:  a) ln .34                 b)  log 2.3                   c) ln (-1.5) d)  4.9 = log x        e)  ln x = - 2.1 Back to Exponenials and Logarithms Main Page Back to the Survey of Math Ideas Home Page