LOGARITHMS   The inverse of the exponential function--  The Natural Logarithm The graph of          y  =  ex  clearly shows that it is a one to one function, hence an inverse exists.  We call this inverse the natural logarithm.  and write it as         y  =  ln x   Below is the graph of          y = ex   and          y   =  ln x  By definition, they are reflections of each other across the line y  =  x.              Inverse Properties of Logs Since logs and exponents cancel each other we have:         eln x  =  x  and             ln ex  =  x   Example         eln 3  =  3            and            ln(e5)  =  5 Three Properties of Logs         Property 1:  ln (uv)  = ln u  +  ln v    (The Product to Sum Rule)         Property 2:  ln (u/v) = ln u  -  ln v    (The Quotient to Difference Rule)         Property 3:  ln ur   =  r ln u                    (The Power Rule)            Example: Expand:           ln (xy2/z) by property 2 we have:         ln (xy2) -  ln z by property 1 we have         ln x + ln y2  - ln z By property 3 we have         ln x + 2 ln y - ln z   Exercise Try to expand                            Example Write as a single logarithm:         4ln x - 1/2ln y  + ln z  Solution We first use property 3 to write:         ln x4 - ln y1/2  + ln z Now we use property 2:                 x4           ln               +  ln z               y1/2   Finally, we use property 3:                 x4 z          ln               +  ln z               y1/2   Exercise Write the following as a single logarithm:         1/3 ln x  +  2 ln y   -  3 ln z         Example Suppose that           ln 3 = 1.10  and that          ln 5 = 1.61 Find          ln 45   Solution Since          45 = (5)(32) We have         ln 45  =  ln (5)(32)  =  ln 5 + ln 32         =   1.61 + 2 ln 3         =  1.61 + 2 (1.10)  =  3.81   Exponential Equations The key to solving equations is to know how to apply the inverse of a function.  When we have an exponential equation, we will use the natural logarithm to cancel the exponential.   Example Find k if         34  =  e10k   Solution Take the natural logarithm of both sides         ln 34  =  ln e10k Now use the inverse property          ln 34  =  10k Finally divide by 10            ln 34                           =  k  =  0.3526              10 Carbon Dating   All living beings have a certain amount of radioactive carbon C14 in their bodies.  When the being dies the C14 slowly decays with a half life of about 5600 years.  Suppose a skeleton is found in Tahoe that has 42% of the original C14.   When did the person die? Solution We can use the exponential decay equation:         y  =  Cekt  After 5600 years there is          C/2  C14 left.  Substituting, we get:         C/2  =  Cek(5600) Dividing by C,         1/2  =  e5600k Take ln of both sides,         ln(0.5)  =  5600k so that                    ln(0.5)         k  =                 =  -0.000124                    5600 The equation becomes         y  =  Ce-0.000124t  To find out when the person died, substitute          y  =  0.42C  and solve for t:         0.42C  =  Ce-0.000124t Divide by C,         0.42  =  e-0.000124t Take ln of both sides,         ln(0.42)  =  -0.000124t Divide by -0.000124                  ln(0.42)         t =                     = 6995                -0.000124 The person died about 7,000 years ago.   Back to the Math Department Home