Tutorials on how to solve exponential and logarithmic equations with examples and detailed solutions are presented. A tutorials with exercises and solutions on the use of the rules of logarithms and exponentials may be useful before you start the present tutorial.
Examples with Solutions
Example 1
ln (x) = 5 and check the solution found.Solution to Example 1 Use the inverse property (9) given above to rewrite the given logarithmic (ln has base e) equation as follows: x = e ^{5} Check Solution Substitute x by e ^{5} in the left side of the given equation and simplify ln (e ^{5}) = 5 , use property (4) to simplify which is equal to the right side. Hence x = e ^{5} is the solution to the given equation.
Example 2
e^{ x} = 6 and check the solution found.Solution to Example 2 Use the inverse property (9) given above to rewrite the given exponential equation as follows: x = ln(6) Check Solution Substitute x by ln(6) in the left hand side of the equation e^{ x} and simplify e^{ ln(6)} = 6 , use property (5) to simplify which is equal to the right hand side of the given equation. Hence x = ln(6) is the solution to the given equation.
Example 3
ln (x) + ln (2) = 3 Solution to Example 3 Use property (1) from right to left to group the two ln terms on the left ln(2 x) = 3 Use property (9) to rewrite the above logarithmic equation as follows: 2 x = e ^{3} Solve for x x = e ^{3} / 2
Example 4
e^{ 3x} = - 1 Solution to Example 4 The range of basic exponential functions is (0 , + ∞), hence e^{ 3x} cannot be negative and therefore the given equation has no real solutions.
Example 5
Solve the logarithmic equation and check the solution obtained.
ln(x) + 2 = - 3 ln (x) + 10 Solution to example 5 combine like terms 4 ln(x) = 8 Divide both sides by 4. ln(x) = 2 Use the inverse property (9) to rewrite the above using exponentials. x = e ^{2} check solution obtained Evaluate left side of the given equation: ln(e ^{2}) + 2 = 2 + 2 = 4 Evaluate right side of equation: - 3 ln(e ^{2}) + 10 = -3(2) + 10 = 4 Hence, the solution to the above equation is x = e ^{2}.
Example 6
2 e ^{x} + e ^{ - x} = 3 Solution to Example 6multiply all terms by e ^{x} (2 e ^{x} + e ^{ - x})e ^{x} = 3e ^{x} Expand 2 e ^{x} e ^{x} + e ^{ - x} e ^{x} = 3 e ^{x} Use exponents properties to simplify. 2 e ^{2 x} + 1 = 3 e ^{x} note that e ^{2 x} = (e ^{x}) ^{2} The equation may be written as 2 (e ^{x}) ^{2} + 1 = 3 e ^{x} Use substitution: let u = e ^{x} and rewrite the equation in u 2 u ^{2} + 1 = 3u rewrite the equation in standard form 2 u ^{2} - 3 u + 1 = 0 solve, for u, the above quadratic equation u = 1 , u = 1/2 We substitute u by e ^{x} in the above solutions e ^{x} = 1 and e ^{x} = 1/2 Solve, for x, the first of the above equations e ^{x} = 1 Use inverse property (9) to rewrite the above in logarithm form (or take logarithms of both sides) x = ln(1) = 0 Solve, for x, the second of the above equations e ^{x} = 1/2 Use inverse property (9) to rewrite the above in logarithm form (or take logarithms of both sides) x = ln(1/2) = - ln(2) check Check first solution: x = 0 Evaluate Left Side of equation 2 e ^{0} + e ^{0} = 2*1 + 1 =3 Right Side of Equation = 3 Check second solution: x = - ln(2) Evaluate Left Side of equation: 2 e ^{- ln(2)} + e ^{-(-ln(2))} = 2/e ^{ln(2)} + e ^{ln(2)} = = 2/2 + 2 = 3 Right Side of Equation = 3 Conclusion The solutions to the given equation are: x = 0 and x = - ln(2).
Example 7
ln (x + 1) + ln (x) = ln (2) Solution to Example 7Use property (1) above to group the two terms on the left side of the equation ln[ (x + 1) x ] = ln(2) Use property (6) to write the algebraic equation (x + 1) x = 2 Expand and write in standard form x ^{2} + x - 2 = 0 Solve the above quadratic equation for x to find the solution x = 1 and x = - 2 check Check first solution: x = 1 Left side: ln (1 + 1) + ln (1) = ln (2) , left side is equal to right side ln(2), x = 1 is a solution Check second solution: x = - 2 left side: ln (-2 + 1) + ln (1) = ln ( -1) + ln(1) , ln(-1) is not defined in the real numbers, hence , x = -2 is NOT a solution to the given equation. Conclusion The given equation has one solution: x = 1
Example 8
ln (x + 1) - ln (x) = 2 Solution to Example 8Use property (2) from right to left to group the term on the left side of the equation ln( (x + 1) / x) = 2 Use property (9) to rewrite the above logarithmic equation as follows: (x + 1) / x = e^{ 2} Cross multiply x + 1 = e^{ 2} x Group terms with x on the left and the constant terms on the right x - e^{ 2} x = - 1 factor x out x(1 - e^{ 2}) = - 1 Solve for x-fast x = 1 / (e^{ 2} - 1) Check the solution found as an exercise.
Example 9
e^{ 2x} e^{ 3x} - 3 = 2 Solution to Example 9 Use property (7) to group the exponential terms on the left e^{ 2 x + 3 x } - 3 = 2 Simplify and rewrite the equation as follows e ^{5 x } = 5 Use property (9) to rewrite the above exponential equation as follows: 5 x = ln (5) x = ln(5) / 5
Example 10
ln (x^{ 4}) + ln (x^{ 2}) - ln (x^{ 3}) - 2 = 7 Solution to Example 10Use properties (1) and (2) to rewrite the left side as ln (x^{ 4} x^{ 2} / x^{ 3}) - 2 = 7 Simplify the expression inside th ln add 2 to both sides of the equation and simplify ln (x^{ 3}) = 9 Use property (9) to rewrite the equation as follows x^{ 3} = e^{9} Solve for x x = ∛(e^{9}) = (e^{9})^{1/3} = e^{3} More References and LinksRules of Logarithms and ExponentialsConvert Logarithms and Exponentials |