MathematicsChapter 2: Logarithms
What A Logarithm Means
A logarithm is an exponent written in a different seat.
Learning objectives
- Explain What A Logarithm Means in simple mathematical language.
- Apply exponent form and log form correctly.
- Apply allowed bases and arguments correctly.
- Use the symbols, conditions, and formulae accurately.
- Solve a short example and check the answer.
- Recognise and correct common mistakes.
Definition
A logarithm is an exponent written in a different seat. By definition, logb x = y if and only if by = x, with base b > 0, b != 1, and argument x > 0. So 25 = 32 is the same fact as log2 32 = 5. You are asking: to what power must I raise b to get x? That question is why logs shrink huge scientific-notation numbers into small exponents you can add.
Formula
x = y if and only if by
Where / symbols
- log_b x = y
- By definition, log_b x = y if and only if b^y = x, with base b > 0, b !
- b^y = x
- By definition, log_b x = y if and only if b^y = x, with base b > 0, b !
- b > 0
- By definition, log_b x = y if and only if b^y = x, with base b > 0, b !
- b != 1
- A key idea used when studying What A Logarithm Means.
- x > 0
- = 1, and argument x > 0.
- 2^5 = 32
- So 2^5 = 32 is the same fact as log_2 32 = 5.
- log_2 32 = 5
- So 2^5 = 32 is the same fact as log_2 32 = 5.
- Base 1
- Base 1 is undefined.
- negative
- = 1 because 1^y is always 1, so it cannot hit other x, and a negative base does not give a clean real power for every y.
- 0
- By definition, log_b x = y if and only if b^y = x, with base b > 0, b !
Relationship
- logb x = y
- by = x
- b != 1
- 25 = 32
Overview
Read the definition both ways, then lock the restrictions:
Exponent Form and Log Form
Start from by = x and name y as log_b x. The base of the power becomes the base of the log. The exponent becomes the log value. The result of the power becomes the argument. Practice flipping: 10^3 = 1000 means log_10 1000 = 3, and 25 = 32 means log2 32 = 5.
Examples 2^5 = 32 <=> log_2 32 = 5 · 10^3 = 1000 <=> log_10 1000 = 3 · 5^2 = 25 <=> log_5 25 = 2
Allowed Bases and Arguments
The base must satisfy b > 0 and b != 1 because 1^y is always 1, so it cannot hit other x, and a negative base does not give a clean real power for every y. The argument must satisfy x > 0 because a positive base raised to a real power stays positive. Logs of zero or negatives are undefined in the reals.
Examples log_2 8 = 3 (allowed) · log_1 5 undefined (base 1) · log_10 (-3) undefined (negative)
WHAT — Core idea
A logarithm is an exponent written in a different seat. By definition, logb x = y if and only if by = x, with base b > 0, b != 1, and argument x > 0. So 25 = 32 is the same fact as log2 32 = 5. You are asking: to what power must I raise b to get x? That question is why logs shrink huge scientific-notation numbers into small exponents you can add.
WHY — Why it matters
The iff (if and only if) is two directions: if the log equals y then the power equals x, and if the power equals x then the log equals y. Always state the three conditions b > 0, b != 1, x > 0 when you define a logarithm.
HOW — Method and examples
1. EXPONENT FORM AND LOG FORM: Start from by = x and name y as log_b x. 2. ALLOWED BASES AND ARGUMENTS: The base must satisfy b > 0 and b !
HOW — Answer in an exam
Base 1 is undefined. Log of a negative or of 0 is undefined in the reals. Do not write log_2 32 = 32; the log is the exponent 5, not the original 32.
Explanation
The iff (if and only if) is two directions: if the log equals y then the power equals x, and if the power equals x then the log equals y. Always state the three conditions b > 0, b != 1, x > 0 when you define a logarithm.
In simple words
Base 1 is undefined. Log of a negative or of 0 is undefined in the reals. Do not write log_2 32 = 32; the log is the exponent 5, not the original 32.
Key points
- logb x = y
- by = x
- b > 0
- b != 1
- log_b x
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.What A Logarithm Means
Sticky Notes · Quick Revision
WHAT
- A logarithm is an exponent written in a different seat.
Key Points
- A logarithm is an exponent written in a different seat.
- Start from by = x and name y as log_b x.
- The base must satisfy b > 0 and b !
- The iff (if and only if) is two directions: if the log equals y then the power equals x, and if the power equals x then the log equals y.
HOW
- 1. EXPONENT FORM AND LOG FORM: Start from by = x and name y as log_b x.
- 2. ALLOWED BASES AND ARGUMENTS: The base must satisfy b > 0 and b !
Examples
- 25 = 32 <=> log2 32 = 5
- 103 = 1000 <=> log10 1000 = 3
- 52 = 25 <=> log5 25 = 2
- log2 8 = 3 (allowed)
- log_1 5 undefined (base 1)
Common Mistake
- Correction: Base 1 is undefined.
- Correction: Log of a negative or of 0 is undefined in the reals.
Exam Tip
- Base 1 is undefined. Log of a negative or of 0 is undefined in the reals. Do not write log_2 32 = 32; the log is the exponent 5, not the original 32.
60-second revision
- A logarithm is an exponent written in a different seat.
- Start from b^y = x and name y as log_b x.
- The base must satisfy b > 0 and b !
- The iff (if and only if) is two directions: if the log equals y then the power equals x, and if the power equals x then the log equals y.
- Base 1 is undefined.
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