# Solve Exponential and Logarithmic Equations - Tutorial

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.

## Review of the Properties and Rules of Logarithm and Exponentials

Some of the most important rules of logarithms and exponentials and properties are listed below.-
log
_{b}(U V) = log_{b}U + log_{b}V -
log
_{b}(U / V) = log_{b}U - log_{b}V -
log
_{b}(U^{r}) = r log_{b}U -
log
_{b}(b^{x}) = x -
b
^{logb x}= x , x > 0 -
log
_{b}U = log_{b}V ⇔ U = V -
a
^{x}a^{y}= a^{x + y} -
a
^{x}/ a^{y}= a^{x - y}

Being inverse of each other, logarithmic and exponential functions to the same base are related as follows: -
log
_{y}x = y ⇔ x = b^{y}

This last inverse function property helps in converting exponential equation to a logarithmic one and a logarithmic equation to an exponential one.

## Examples with Solutions
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.
e 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.
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
e 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.
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}.
2 e multiply 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 u2 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).
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
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.
e 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
ln (x Use 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 |