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Special Log Values
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, log_a (a^x) = 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.
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Special Log Values
What is it?
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, log_a (a^x) = 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.
In simple words
- 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.
Important terms
logb 1 = 0: logb 1 = 0 because b0 = 1.
logb b = 1: logb b = 1 because b1 = b.
loga (a^x) = x: For matching base and argument power, loga (ax) = x (the log undoes the exponential).
logb 1: logb 1 = 0 because b0 = 1.
0: logb 1 = 0 because b0 = 1.
What you should learn
- 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.
Examples
- log10 1 = 0
- log10 10 = 1
- log2 (27) = 7
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, log_a (a^x) = x. These are the same fact as the definition, just specialised.
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 10^y = -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.
Recommended for Class 9
Full school-level explanation
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, log_a (a^x) = 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.
Memorise the three 'free' values, then the bans:
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, log_a (a^x) = 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 (a^x) = x as 'log and exponential with the same base cancel.' The twin identity a^(loga 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 log_b 1 = 0. 2. WHERE LOGS FAIL: Base 1 is undefined: 1^y never equals 8, for example.
HOW — Answer in an exam
logb 1 is 0, not 1. logb 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.
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, log_a (a^x) = x. These are the same fact as the definition, just specialised.
Remember
- logb 1 = 0
- logb b = 1
- loga (ax) = x
Examples
- log10 1 = 0
- log10 10 = 1
- log2 (27) = 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 10^y = -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.
Remember
- Base 1
- undefined
- negative
- 0
Examples
- log_1 10 undefined
- log_10 0 undefined
- log_2 (-8) undefined in reals
Remember
- logb 1 = 0
- logb b = 1
- loga (ax) = x
- loga (ax) = x
- log_b 1
- 0
- log_b b
- 1
How the ideas connect
You can also read loga (a^x) = x as 'log and exponential with the same base cancel.' The twin identity a^(loga x) = x holds for x > 0. Together they say logs and powers undo each other.
Deeper understanding
Go beyond memorising Special Log Values. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
You can also read loga (a^x) = x as 'log and exponential with the same base cancel.' The twin identity a^(loga x) = x holds for x > 0. Together they say logs and powers undo each other.
Common mistakes
Exam focus
WHAT
Key Points
HOW
Examples
Exam focus
Quick revision
- logb 1 = 0
- logb b = 1
- loga (ax) = x
- logb 1 = 0: logb 1 = 0 because b0 = 1.
- logb b = 1: logb b = 1 because b1 = b.
- loga (a^x) = x: For matching base and argument power, loga (ax) = x (the log undoes the exponential).
logb 1 is 0, not 1. logb 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.
For long answers on Special Log Values: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- logb 1 = 0
- logb b = 1
- loga (ax) = x
- logb 1 = 0: logb 1 = 0 because b0 = 1.
Important formulas
- logb 1 = 0
- logb b = 1
- loga (ax) = x
Common exam traps
- log_b 1 is 0, not 1.
- log_b b is 1, not b.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
log_b 1 is 0, not 1.
- Mistake
log_b b is 1, not b.
- Mistake
Students mix those two constantly.
- Mistake
Also, log_10 0 is not 0; 10^0 is 1, so the log of 1 is 0.
- Mistake
logb 1 is 0, not 1. logb 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.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
logb 1 = 0
logb b = 1
loga (ax) = x
logb 1 = 0: logb 1 = 0 because b0 = 1.
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, log_a (a^x) = x (the
logb 1 = 0
- 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, log_a (a^x) = x (the
- Formula
- logb 1 = 0
- Example
- log10 1 = 0
- Critical warning
- log_b 1 is 0, not 1.
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Exam Practice
Define
Q1. Define Special Log Values.
Explain
Q2. Explain: THREE STANDARD RESULTS.
Example
Q3. Give one clear example of Special Log Values and say why it fits.
Mistake
Q4. State one common mistake students make about Special Log Values.
Exam note
Q5. Write a short exam note on Special Log Values: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.