Solve Exponential and Logarithmic Equations - Tutorial

Solve Exponential and Logarithmic Equations - Tutorial

How to solve exponential and logarithmic equations? A tutorial with examples and detailed solutions. Logarithmic equations are equations with logarithmic terms and exponential equations are equations with exponential terms.

Example 1: Solve the logarithmic equation.

ln(x) + 2 = - 3ln(x) + 10
Solution to example 1
  • combine like terms
      4ln(x) = 8


  • divide both sides by 4.
      ln(x) = 2


  • exponentiate both sides.
      eln(x) = e2


  • inverse property of exponents and logs
      x = e2


  • check:
      Left Side of equation:
      ln(e2) + 2
      = 2 + 2
      = 4
      Right Side of equation:
      - 3ln(e2) + 10
      = -3(2) + 10
      = 4


  • conclusion: The solution to the above equation is
      x = e2.



Example 2: Solve the exponential equation.
2ex + e-x = 3
Solution to example 2
  • multiply all terms by ex
      (2ex + e-x)ex = 3ex


  • use exponents properties to simplify.
      2e2x + 1 = 3ex



  • note that.
      e2x = (ex)2


  • let u = ex and rewrite the equation in u
      2u2 + 1 = 3u


  • rewrite the equation
      2u2 - 3u + 1 = 0


  • solve, for u, the above quadratic equation
      u = 1 , u = 1/2


  • but u = ex
      ex = 1
      ex = 1/2


  • solve, for x, the first of the above equations
      ex = 1


  • take logarithms of both sides
      ln(ex) = ln1


  • which gives
      x = 0


  • solve, for x, the second of the above equations
      ex = 1/2


  • take logarithms of both sides
      ln(ex) = ln(1/2)


  • which gives
      x = -ln2


  • check:
      1st solution x = 0
      Left Side of equation: 2e0+e0 = 2*1 + 1 =3
      Right Side of Equation: 3

      2nd solution x = -ln2
      Left Side of equation: 2e-ln2 + e-(-ln2)
      = 2/eln2 + eln2 =
      = 2/2 + 2 = 3
      Right Side of Equation: 3

conclusion: The solutions to the given equation are
x = 0 and x = -ln2.



Take a self test on exponential and logarithmic functions..


More topics to explore and Tests:
Experiment and Explore Mathematics: Tutorials and Problems

More To Explore