Log Logic Presentation Log Logic
Cracking the Logarithmic Code
The Golden Rule
To solve a logarithmic equation, we must speak its language.
\[ \log_{b}(x) = y \iff b^{y} = x \]
b Base
y Power
x Result
The "Loop" Method
Start at the base and travel in a circle:
1
Base \( b \)
2
To the power of \( y \)
3
Equals \( x \)
\( \log_{b}(x) = y \)
\( b^{y} = x \)
The Blueprint
01. Isolate
Get the log term by itself.
02. Convert
Rewrite in exponential form.
03. Solve
Solve for the variable.
04. Verify
Check: Argument must be positive.
Example 01
Standard Form
\[ \log_{2}(x) = 5 \]
Exponential Form:
\[ 2^{5} = x \]
Answer: \( x = 32 \)
Example 02
Expression Arguments
\[ \log_{3}(x - 4) = 2 \]
Step 1: Rewrite
\[ 3^{2} = x - 4 \]
Step 2: Solve
\[ 9 = x - 4 \implies x = 13 \]
Example 03
Isolating First
\[ 2\log_{5}(x) = 4 \]
1. Divide
\( \log_{5}(x) = 2 \)
2. Convert
\( 5^{2} = x \)
3. Solve
\( x = 25 \)
Warning!
Extraneous Solutions
The argument of a log must be positive.
\( \log_{b}(\text{Result}) \)
Always check your final answer in the original log!
Log Logic Notes Worksheet Log Logic
Solving Logarithmic Equations
Name:
Date:
The Key Concept
\( \log_{b}(x) = y \)
\( b^{y} = x \)
b is the...
y is the...
x is the...
The Blueprint Steps
Step 1
Step 2
Step 3
Step 4
Guided Examples
Example 1 \( \log_{2}(x) = 5 \)
Rewrite...
Solve...
Example 2 \( \log_{3}(x - 4) = 2 \)
Rewrite...
Solve...
Example 3 \( 2\log_{5}(x) = 4 \)
Isolate...
Solve...
Independent Mission
01 \( \log_{4}(x) = 3 \)
02 \( \log_{2}(x + 1) = 4 \)
03 \( 3\log_{2}(x) = 15 \)
04 \( \log_{10}(2x) = 2 \)
The Extraneous Check
Check the original logarithm argument!
Argument > 0
Log Logic Answer Key Answer Key
Material: Log Logic Notes
Teacher Resource
Key Concepts & Steps
Base
b
Exponent
y
Argument
x
1. ISOLATE
2. CONVERT
3. SOLVE
4. CHECK
Guided Examples
Ex 1: \( \log_{2}(x) = 5 \)
\( 2^{5} = x \) \( x = 32 \)
Ex 2: \( \log_{3}(x - 4) = 2 \)
\( 3^{2} = x - 4 \) \( x = 13 \)
Ex 3: \( 2\log_{5}(x) = 4 \)
\( \log_{5}(x) = 2 \) \( x = 25 \)
Independent Mission Key
Prob 01 \( \log_{4}(x) = 3 \)
\( 4^{3} = x \)
\( x = 64 \)
Prob 02 \( \log_{2}(x + 1) = 4 \)
\( 2^{4} = x + 1 \implies 16 = x + 1 \)
\( x = 15 \)
Prob 03 \( 3\log_{2}(x) = 15 \)
Isolate log first:
\( \log_{2}(x) = 5 \implies 2^{5} = x \)
\( x = 32 \)
Prob 04 \( \log_{10}(2x) = 2 \)
\( 10^{2} = 2x \implies 100 = 2x \)
\( x = 50 \)
Log Logic Practice Worksheet Log Logic Practice
Solving Logarithmic Equations
Name:
Date:
Level 01
Basic Conversions
Question 01
\[ \log_{5}(x) = 2 \]
Question 02
\[ \log_{2}(x) = 6 \]
Question 03
\[ \log_{10}(x) = 3 \]
Question 04
\[ \log_{4}(x) = 0 \]
Question 05
\[ \log_{3}(x) = -2 \]
Question 06
\[ \log_{7}(x) = 1 \]
Level 02
Expressions
Question 07
\[ \log_{3}(x+2) = 4 \]
Question 08
\[ \log_{2}(x-5) = 3 \]
Question 09
\[ \log_{5}(2x) = 2 \]
Question 10
\[ \log_{4}(3x - 1) = 2 \]
Question 11
\[ \log_{4}(x + 12) = 3 \]
Question 12
\[ \log_{2}\left(\frac{x}{4}\right) = 5 \]
Level 03
Isolation
Question 13
\[ 2\log_{4}(x) = 6 \]
Question 14
\[ \log_{3}(x) + 5 = 7 \]
Question 15
\[ -3\log_{2}(x) = -12 \]
Question 16
\[ 5 + \log_{10}(x) = 2 \]
Question 17
\[ 4\log_{2}(x-1) = 12 \]
Question 18
\[ \log_{5}(2x+10) = 3 \]
Question 19
\[ \frac{1}{2}\log_{9}(x) = 1 \]
Question 20
\[ 10 - \log_{2}(x) = 8 \]
Verification: Original log argument must be positive.
Log Logic Practice Key Answer Key
Material: Log Logic Practice
Teacher Resource
Questions 01 - 10
01
\( 5^{2} = x \) x = 25
02
\( 2^{6} = x \) x = 64
03
\( 10^{3} = x \) x = 1000
04
\( 4^{0} = x \) x = 1
05
\( 3^{-2} = x \) x = 1/9
06
\( 7^{1} = x \) x = 7
07
\( 81 = x + 2 \) x = 79
08
\( 8 = x - 5 \) x = 13
09
\( 25 = 2x \) x = 12.5
10
\( 16 = 3x - 1 \) x = 17/3
Questions 11 - 20
11
\( 64 = x + 12 \) x = 52
12
\( 32 = x/4 \) x = 128
13
\( \log_4(x) = 3 \) x = 64
14
\( \log_3(x) = 2 \) x = 9
15
\( \log_2(x) = 4 \) x = 16
16
\( \log_{10}(x) = -3 \) x = 0.001
17
\( \log_2(x-1) = 3 \) x = 9
18
\( \log_5(2x+10) = 3 \) x = 57.5
19
\( \log_9(x) = 2 \) x = 81
20
\( \log_2(x) = 2 \) x = 4
