Solving Exponential & Logarithmic Equations
Properties of Exponential and Logarithmic Equations
Let be a positive real number such that 1, and let and be real numbers. Then the following properties are
true:
1.
if and only if
2. log
log
if and only if 0, 0
Inverse Properties of Exponents and Logarithms
Base a Natural Base e
1. log
ln
2.
Solving Exponential and Logarithmic Equations
1. To solve an exponential equation, first isolate the exponential expression, then take the logarithm of both sides
of the equation and solve for the variable.
2. To solve a logarithmic equation, first isolate the logarithmic expression, then exponentiate both sides of the
equation and solve for the variable.
For Instance: If you wish to solve the equation, ln 2, you exponentiate both sides of the equation to solve it
as follows:
ln 2 Original equation
Exponentiate both sides
Inverse property
Or you can simply rewrite the logarithmic equation in exponential form to solve (i.e. ln 2 if and only if
).
Note: You should always check your solution in the original equation.
Example 1:
Solve each equation.
a. 4
"
64 b. ln2 $ 3 ln 11
Solution:
a. 4
"
64 Original Equation
4
"
4
&
Rewrite with like bases
' 2 3 Property of exponential equations
1 Subtract 2 from both sides
The solution is 1. Check this in the original equation.
b. ln2 $ 3 ln 11 Original Equation
2 $ 3 11 Property of logarithmic equations
2 14 Add 3 to both sides
7 Divide both sides by 2
The solution is 7. Check this in the original equation.
Example 2:
Solve 5 '
"*
20.
Solution:
5 '
"*
20 Original Equation
"*
15 Subtract 5 from both sides
ln
"*
ln 15 Take the logarithm of both sides
' 1 ln 15 Inverse Property
$ 1 ' ln 15 + 1.708 Subtract 1 from both sides
Check:
5 '
"*
20 Original Equation
5 '
*../0"*
20 Substitute 1.708 for
5 '
../0
20 Simplify
5 ' 14.999 + 20 Solution checks
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