# 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