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Solving Exponential & Logarithmic Equations
Properties of Exponential and Logarithmic Equations
Let be a positive real number such that , and let and be real numbers. Then the following properties are
true:
1.
2.
Inverse Properties of Exponents and Logarithms
Base a Natural Base e
1.
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,
, you exponentiate both sides of the equation to solve it
as follows:
Original equation
Exponentiate both sides
Inverse property
Or you can simply rewrite the logarithmic equation in exponential form to solve (i.e.
).
Note: You should always check your solution in the original equation.
Example 1:
Solve each equation.
"
a. ! #! b.
$%
Solution:
"
a. ! #! Original Equation b.
$%
Original Equation
!" !& Rewrite with like bases $ % Property of logarithmic equations
' % Property of exponential equations ! Add 3 to both sides
Subtract 2 from both sides ( Divide both sides by 2
The solution is 1. Check this in the original equation. The solution is 7. Check this in the original equation.
Example 2:
Solve ) ' "* .
Solution: Check:
) ' "* Original Equation ) ' "* Original Equation
"* *,./0"* ?
) Subtract 5 from both sides )' Substitute 1.708 for
"* ,./0 ?
) Take the logarithm of both sides ) ' Simplify
'
) Inverse Property ) ' !,111 + Solution checks
$'
)+,(- Subtract 1 from both sides
Example 3:
Solve the exponential equations.
a. ( b. !2& 1 c.
Solutions:
Method 1: Method 2:
a. ( Original Equation a. ( Original Equation
( Take the logarithm of both sides ( Take the logarithm of both sides
( Property of Logarithms ( Inverse Property
. + ,-( Solve for . + ,-( Change of Base Formula
2& Method 1: 2& Method 2:
b. ! 1 Original Equation b. ! 1 Original Equation
!2& 1 Take the logarithm of both sides 4 !2& 41 Take the logarithm of both sides
$% !1 Property of Logarithms $ % 4 1 Inverse Property
$% 3 Divide both sides by ! $ % 3 Change of Base Formula
4 4
%' 3 + !,)-) Solve for %' 3 + !,)-) Solve for
4 4
c. Original Equation
) Divide both sides by 2
) Take the logarithm of both sides
) + ,#1 Inverse Property
Example 4: Example 5:
Solve 4 ). Solve % #.
Solution: Solution:
4 ) Original Equation % # Original Equation
4 5 Divide both sides by 2 Divide both sides by 3
6 Change to exponential form
!5 Change to exponential form Simplify
% Simplify
Example 6: Example 7: Solving a Logarithmic Equation
Solve
, %. using Exponentiation
Solution: Solve & $ & $ %
, % Original Equation Solution:
, ,) Divide both sides by 20 & $& $% Original Equation
, *,5 Change to exponential form & Condense the left side
*,5 2&
) + ,!- Divide both sides by 0.2 89 *
% 79:7 % Exponentiate both sides
% Inverse Property
2&
% $ 1 Multiply both sides by $ %
1 Solve for
Practice Problems
Solve the following equations:
Remember that the arguments of all logarithms must be greater than 0. Also exponentials in the form of
will be greater than 0. Be sure to check all your answers in the original equation.
2* 2
1. % - 22. % -
2. - ! 23. & )
3. ) 24. 4 %
4. $!'% 25.
5. $#'
% 26.
' !
(
6. % ' 27. & '
7.
$
% ! 28. 5 $
8.
% ! 29. % )
9. )" ! 30. -
31.
(, )
10.
' #
2& 32.
$,)
11. ! , ) /,5
12. $ ) $% 33. !)
13. . % ' . . % 34. 2/,;
14. ;! 35. $! -
<
15. ' $ % 36. ) $
16. ' ) $ $ % 37. ,)
17. !
'% 38. $,#)
18. $# ! *
39. ') (
19. #! &
20. ) ) 40. !5 ' !,-
21. !2& * 41. ' % %
*; 42. 4 $4 $
Practice Problems Answers
1. 5 22. 6
23. 243
2.
& 24. 64
3. 1.609 25. 50
4. 2.120 26. 3
5. 134.476 27. 4
6. 33 28. 35
7. 163.794 29. 5.66
8. 2.463 30. 3.32
9. -1.139 31. 1408.10
10. 18.086, -22.086 32. 0.61
*
11.
& 33. 6.23
12. 1.099 34. 2.68
&
13. & 35. No Solution
*
14.
4 36. –0.65
15. 4 37. 15.81
16. 3 38. 0.32
17. 6.321 39. 64
18. 96 40. 5.90
19. 6 0
41.
20. 2 &
21. 1 42. 2
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