MHF4U_2011: Advanced Functions, Grade 12, University Preparation
Unit 3: Exponential and Logarithmic Functions
Activity 5: Solving Logarithmic Equations
Homework/Formative Assessment
1. Put the following sequence of steps in the correct order:
4x + 50 = 100 4x + 50 = 102 log10(4x + 50) = 2 x = 25/2
4x = 50
2. Solve the following logarithmic equations and check your answers.
a) log2(x - 4) = 4 b) log(2x - 3) = 1
c) log5(3x + 8) = 2 d) logx(9) = 2
e) log2(x + 2) = - 2 f) log3(4x - 7) = 3
3. Explain why no solutions exist for the following equation: log-327 = x.
4. Solve for x in the following equation:
log3x - log36 = log318 - log32 5. Solve log2(x - 5) + log2(x - 2) = 2 6. Estimate the value of the following:
a) log2200 b) log319 c) log4150
Homework/Formative Assessment - SOLUTIONS
1. Put the following sequence of steps in the correct order:
4x + 50 = 100 log10(4x + 50) = 2
4x + 50 = 102 4x + 50 = 102
log10(4x + 50) = 2 4x + 50 = 100
x = 25/2 4x = 50
4x = 50 x = 25/2
2. Solve the following logarithmic equations and check your answers.
a) log2(x - 4) = 4 4 24
16 4 20 x x x
− =
= +
=
b) log(2x - 3) = 1 2 3 101
2 10 3 13
2 x x x
− =
= +
= c) log5(3x + 8) = 2
3 8 52
3 25 8
17 3 x x x
+ =
= −
=
d) logx(9) = 2 9 2
9 3 3
3 (-3 is not in domain) x
x x
=
± =
± =
= e) log2(x + 2) = - 2
2 2 2
1 2 4
7 4 x x x
+ = −
= −
= −
f) log3(4x - 7) = 3 4 7 33
4 27 7
34 4 17
2 x x x x
− =
= +
=
=
3. Explain why no solutions exist for the following equation: log-327 = x.
Written in exponential form, you have ( 3)− x=27. There is no exponent that you can raise -3 (a negative) to that will give a result of 27 (a positive).
4. Solve for x in the following equation:
log3x - log36 = log318 - log32
( )
3 3 3 3
3 3 3 3
3 3
3 3
log x log 6 log 18 log 2 log x log 18 log 2 log 6 log x log 18 6
2 log x log 54
54 x
− = −
= − +
= ×
=
=
5. Solve log2(x - 5) + log2(x - 2) = 2
( ) ( )
( ) ( )
( )
( ) ( )
2 2
2 2 2
2 2
2 2 2
log x 5 log x 2 2
log 5 2 2
log 7 10 2
x 7 10 2
x 7 10 4
x 7 10 4 0
x 7 6 0
6 1 0
6 or 1
6 ( 1 is not in the domain of
this function, since if you put it back into the input ( - 5) o
x x
x x
x x x x
x x
x x
x x
x
− + − =
− − =
− + =
− + =
− + =
− + − =
− + =
− − =
= =
= =
r ( - 2) you will get a negative )x 6. Estimate the value of the following:
a) log2200 2 200
7.64
x
x
=
≈
b) log319 3 19
2.68
x
x
=
≈
c) log4150 4 150
3.61
x
x
=
≈
