# 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.
1. logb(U V) = logb U + logb V
2. logb(U / V) = logb U - logb V
3. logb (Ur) = r logb U
4. logb (bx) = x
5. blogb x = x , x > 0
6. logb U = logb V ? U = V
7. a x a y = a x + y
8. a x / a y = a x - y
Being inverse of each other, logarithmic and exponential functions to the same base are related as follows:
9. logy 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

Example 1
Solve the equation

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
Find all real solutions to the exponential equation

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
Solve the 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

Example 4
Find all real solutions to the equation

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
Solve the equation and check the solution found.

2 e x + e - x = 3

Solution to Example 6
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 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
Find all real solutions to the equation

ln (x + 1) + ln (x) = ln (2)

Solution to Example 7
Use 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
Find all real solutions to the equation

ln (x + 1) - ln (x) = 2

Solution to Example 8
Use 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
Find all real solutions to the equation

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
Find all real solutions to the equation

ln (x 4) + ln (x 2) - ln (x 3) - 2 = 7

Solution to Example 10
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 = e9
Solve for x
x = ∛(e9) = (e9)1/3 = e3