MathematicsChapter 2: Logarithms
Special Log Values
A few identities save time in every paper.
Learning objectives
- Explain Special Log Values in simple mathematical language.
- Apply three standard results correctly.
- Apply where logs fail correctly.
- Use the symbols, conditions, and formulae accurately.
- Solve a short example and check the answer.
- Recognise and correct common mistakes.
Definition
A few identities save time in every paper. logb 1 = 0 because b0 = 1. logb b = 1 because b^1 = b. For matching base and argument power, loga (ax) = x (the log undoes the exponential). These hold only where the definition allows: b > 0, b != 1, and a > 0, a != 1, with a^x > 0 automatically when a is a valid base.
Formula
b = 1 because b1
Where / symbols
- log_b 1 = 0
- log_b 1 = 0 because b^0 = 1.
- log_b b = 1
- log_b b = 1 because b^1 = b.
- log_a (a^x) = x
- For matching base and argument power, log_a (a^x) = x (the log undoes the exponential).
- log_b 1
- log_b 1 = 0 because b^0 = 1.
- 0
- log_b 1 = 0 because b^0 = 1.
- log_b b
- log_b b = 1 because b^1 = b.
- 1
- log_b 1 = 0 because b^0 = 1.
Relationship
- logb 1 = 0
- logb b = 1
- loga (ax) = x
Overview
Memorise the three 'free' values, then the bans:
Three Standard Results
Anything (allowed) to the power 0 is 1, so logb 1 = 0. The base to the power 1 is itself, so logb b = 1. The logarithm with base a of a^x returns the exponent, loga (ax) = x. These are the same fact as the definition, just specialised.
Examples log_10 1 = 0 · log_10 10 = 1 · log_2 (2^7) = 7
Where Logs Fail
Base 1 is undefined: 1^y never equals 8, for example. A negative argument is undefined in real numbers: there is no real y with 10y = -5. Log of 0 is undefined because 10^y gets closer and closer to 0 but never lands on it for a finite y. Write 'undefined' rather than inventing a number.
Examples log_1 10 undefined · log_10 0 undefined · log_2 (-8) undefined in reals
WHAT — Core idea
A few identities save time in every paper. logb 1 = 0 because b0 = 1. logb b = 1 because b^1 = b. For matching base and argument power, loga (ax) = x (the log undoes the exponential). These hold only where the definition allows: b > 0, b != 1, and a > 0, a != 1, with a^x > 0 automatically when a is a valid base.
WHY — Why it matters
You can also read loga (ax) = x as 'log and exponential with the same base cancel.' The twin identity a^(log_a x) = x holds for x > 0. Together they say logs and powers undo each other.
HOW — Method and examples
1. THREE STANDARD RESULTS: Anything (allowed) to the power 0 is 1, so logb 1 = 0. 2. WHERE LOGS FAIL: Base 1 is undefined: 1^y never equals 8, for example.
HOW — Answer in an exam
log_b 1 is 0, not 1. log_b b is 1, not b. Students mix those two constantly. Also, log_10 0 is not 0; 10^0 is 1, so the log of 1 is 0.
Explanation
You can also read loga (ax) = x as 'log and exponential with the same base cancel.' The twin identity a^(log_a x) = x holds for x > 0. Together they say logs and powers undo each other.
In simple words
log_b 1 is 0, not 1. log_b b is 1, not b. Students mix those two constantly. Also, log_10 0 is not 0; 10^0 is 1, so the log of 1 is 0.
Key points
- logb 1 = 0
- logb b = 1
- loga (ax) = x
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Special Log Values
Sticky Notes · Quick Revision
WHAT
- A few identities save time in every paper.
Key Points
- A few identities save time in every paper.
- Anything (allowed) to the power 0 is 1, so logb 1 = 0.
- Base 1 is undefined: 1^y never equals 8, for example.
- You can also read loga (ax) = x as 'log and exponential with the same base cancel.
HOW
- 1. THREE STANDARD RESULTS: Anything (allowed) to the power 0 is 1, so logb 1 = 0.
- 2. WHERE LOGS FAIL: Base 1 is undefined: 1^y never equals 8, for example.
Examples
- log_10 1 = 0
- log_10 10 = 1
- log2 (27) = 7
- log_1 10 undefined
- log_10 0 undefined
Common Mistake
- log_b 1 is 0, not 1.
- log_b b is 1, not b.
Exam Tip
- log_b 1 is 0, not 1. log_b b is 1, not b. Students mix those two constantly. Also, log_10 0 is not 0; 10^0 is 1, so the log of 1 is 0.
60-second revision
- A few identities save time in every paper.
- Anything (allowed) to the power 0 is 1, so log_b 1 = 0.
- Base 1 is undefined: 1^y never equals 8, for example.
- You can also read log_a (a^x) = x as 'log and exponential with the same base cancel.
- log_b 1 is 0, not 1.
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