Solutions to Exponential and Logarithm Problems

Solution to Problem 1:

• If t = 0 corresponds to 1980 then 1990 corresponds to t = 10. We can say that when t = 10 the population was 10000 and this gives the following equation:
  
  5000 e ^{10 a} = 10000
  
  
• Divide both sides of the equation by 5000 and simplify
  
  e ^{10 a} = 10000 / 5000
  
  e ^{10 a} = 2
  
  
• Using ln, the above exponential form can be written as
  
  10 a = ln 2
  
  
• Solve the above equation for a and approximate the answer to 3 decimal places.
  
  a = ln 2 / 10 = 0.069 (approximated to 3 decimal places).

Solution to Problem 2:

• If t = 0 corresponds to the year 2000 then t = 20 corresponds to the year 2020. We need to find k so that the two populations are equal (P1 = P2) when t = 20.
  
  10000 e ^{20 k} = 20000 e ^{20 * 0.01}
  
  
• Divide both sides of the equation by 2000 e ^{20 k} and simplify
  
  10000 / 20000 = e ^{20 * 0.01} / e ^{20 k}
  
  1 / 2 = e ^{0.2 - 20 k}
  
  
• The above exponential form may be converted in logarithmic form as follows
  
  0.2 - 20 k = ln (1/2)
  
  
• Solve for k
  
  k = [0.2 - ln (1/2)] / 20 = 0.045
  (approximated to 3 decimal places).

Solution to Problem 3:

• If I = 10 ^-8 watts/cm ² then
  
  D = 10 log( I / 10 ^-16 )
  
  = 10 log( 10 ^-8 / 10 ^-16 )
  
  = 10 log (10 ⁸) = 80 decibels.
  
  

Solution to Problem 4:

• Using the decibel formulas from problem 3, we can write
  
  60 = 10 log( I1 / 10 ^-16 )
  and
  80 = 10 log( I2 / 10 ^-16 )
  
  
• Subtract the above equations side by side as follows
  60 - 80 = 10 log( I1 / 10 ^-16 ) - 10 log( I2 / 10 ^-16 )
  
  
• The above may be written
  
  20 = 10 [ log( I2 / 10 ^-16 ) - log( I1 / 10 ^-16 ) ]
  
  
• Divide both sides by 10 and use the logarithm property: log x - log x = log (x / y) to rewrite the above equation as follows
  
  2 = log [I2 / I1]
  
  
• The above gives
  
  I2 / I1 = 10 ² = 100.

Solution to Problem 5:

• If the number of people infected is N = 2000. We use this information to write the following equation:
  
  2000 = 15000 / [ 1 + 100 e ^{-0.5 t} ]
  
  
• To find the number of days we need to solve the above equation for t.
  
  
• Rewrite the above equation as follows
  
  [ 1 + 100 e ^{-0.5 t} ] = 15000 / 2000
  
  
• Which can be written as follows
  
  e ^{-0.5 t} = [ 15000 / 2000 - 1 ] / 100
  
  
• We now use the natural logarithm ln to rewrite the above equation as follows
  
  -0.5 t = ln [ ( 15000 / 2000 - 1 ) / 100 ]
  
  
• We now solve for t
  
  t = ln [ ( 15000 / 2000 - 1 ) / 100 ] / -0.5 = 5.467 (approximated to 3 decimal places).

Solution to Problem 6:

• The half life (T) of a radioactive material, is the period of time after which the amount of this material decays to half its initial amount. The initial amount is found by setting t = 0 in the formula A(t) = A _o e ^{-k t}. Which gives an initial amount equal to A _o. Using the above definition, we can write
  
  A _o e ^{-k T} = A _o / 2.
  
  
• Divide both sides by A _o to obtain
  
  e ^{-k T} = 1/2
  
  
• Use ln to change the exponential form into logarithmic form
  
  -k T = ln(1/2)
  
  
• Simplify ln (1/2) and solve for T.
  
  T = ln 2 / k

Solution to Problem 7:

• Given
  
  y = a / [1 + b e ^{-k t}]
  
  
• Multiply both sides by 1 + b e ^{-k t} and divide both sides by y to obtain
  
  1 + b e ^{-k t} = a / y
  
  
• Subtract 1 to both sides
  b e ^{-k t} = a / y - 1
  
  
• Divide both sides by b
  
  e ^{-k t} = [ a / y - 1 ] / b
  
  
• Rewrite the exponential form into logarithmic form
  
  -k t = ln ([ a / y - 1 ] / b)
  
  
• Solve for t.
  
  t = -(1/k) ln ([ a / y - 1 ] / b)

Solution to Problem 8:

• Use the given information to write the equation
  
  V _o e ^{-a t} = 50% V _o
  
  
• Divide both sides by V _o.
  
  e ^{-a t} = 1/2
  
  
• Use ln to change exponential form into logarithmic form
  
  -a t = ln(1/2)
  
  
• Solve for t
  
  t = ln 2 / a

Solution to Problem 9:

• We substitute the ratio in decibels and Pi in the given formula
  
  10 = 10 Log (10 / Pi)
  
  
• Divide both sides by 10.
  
  1 = Log (10 / Pi)
  
  
• Rewrite the logarithmic form into exponential form
  
  10 / Pi = 10 ¹ = 10
  
  
• Solve for Pi.
  
  Pi = 1 mw

Solution to Problem 10: