Exponential and Logarithmic Equations

When you have completed this section you should be able to:

• Solve exponential equations analytically
• Solve logarithmic equations analytically
• Identify exponential/logarithmic equaitons that cannot be solved analytically and solve them graphically or numerically

Simple Equations

• Exponential

  Given: 35 = 2^3x
  1. First take the logs (or the natural log - ln) of both sides of the equation.

     log(35) = log(2^3x)

  2. Now move the exponent outside the log on the right hand side using the fact that log( a^p) = p log(a)

     log(35) = 3x ( log(2) )

  3. Divide both sides of the equation by 3(log(2))

     x = log(35)/(3 (log(2))

  4. Calculate x using your calculator. x = 1.7098

  Now you try it using Practice Problem 1: 68 = 4^5x (answers at the end of this section)

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• Logarithm

  Given: log(x+3) = 2
  1. Using the relationships log(M) = N and M = 10^N to convert the log equation to the power equation x+3 = 10²
  2. Simplify:

     • x + 3 = 100
     • x = 100 - 3
     • x = 97

Now you try it using Practice Problem 2: log(x² - 8) = 0 (answers at the end of this section)

More Complicated Equations

Exponential

Given: 3^x = 4^(x+1)
• First note that both exponents involve the variable (this is good news) Take the log (or ln ) of both sides of the equation.

  log(3^x ) = log( 4^(x+1))

• Now using the property of logs that says log( a^p) = p log(a) pull the exponents out of the log( )

  x log(3) = (x+1) log (4)

• Perform the multiplication called for on the right side of the equation.

  x log(3) = x log(4) + 1 log(4)

• Move x log(4) to the left hand side of the equation.

  x log(3) - x log(4) = log(4)

• Factor the x (it is a common term on the left side of the equation)

  x [log(3) - log(4)] = log(4)

• Divide both sides of the equation by [log(3) - log(4)]

  x = log(4)/[log(3) - log(4)]

• Now calculate the result, x = -4.8188

  And here is Practice Problem 3: 5.1^(x+2) = 8.7^{(x -1)}

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  Practice Problem Answers:

  Practice Problem 1: 68 = 4^5x
  • log( 68 ) = log( 4^5x)
  • log( 68 ) = 5x log( 4 )
  • x = log( 68 )/(5 log(4))
  • x = 0.60875
  Practice Problem 2: log(x² - 8) = 0
  • x² - 8 = 10⁰
  • x² - 8 = 1
  • x² = 1 + 8 = 9
  • x = + 3 AND X = -3 (do both roots work in the original equation?)
  Practice Problem 3: 5.1^x+2 = 8.7^{(x -1)}
  • log (5.1^x+2 ) = log (8.7^{(x -1)})
  • (x+2) log(5.1) = (x - 1) log(8.7)
  • x log(5.1) + 2 log (5.1) = x log(8.7) - log(8.7); Move "like" terms to the same side of the equation.
  • x log(5.1) - x log(8.7) = -2log(5.1) - log(8.7), Now factor the x from the left hand side of the equation.
  • x [log(5.1) - log(8.7)] = -2log(5.1) - log(8.7), divide by the coefficient of x which is [log(5.1) - log(8.7)]
  • x = [-2log(5.1)-log(8.7)]/[log(5.1) - log(8.7)]
  • x = 10.1516