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Logarithms
Mathematical Definitions · Properties · Exponent Connections
1 · What is a Logarithm?
A logarithm answers the question: "To what power must we raise a base to obtain a given number?"
It is the inverse operation of exponentiation — just as subtraction undoes addition, taking a logarithm
undoes raising a base to a power.
If by^ = x (b > 0, b ≠ 1, x > 0) then logb(x) = y
Here b is the base , x is the argument (the number we are taking the log of), and y is the exponent
(the result of the logarithm). The exponent form and the logarithm form are two ways of writing exactly
the same relationship.
Exponent Form Logarithm Form What it means
23 = 8 log 2 (8) = 3 2 raised to 3 gives 8
102 = 100 log 10 (100) = 2 10 raised to 2 gives 100
50 = 1 log 5 (1) = 0 Any base raised to 0 gives 1
3 -1^ = 1/3 log 3 (1/3) = − 1 Negative exponent → fraction
e^1 ≈ 2.718 ln(e) = 1 Natural log base e
Key insight: every logarithm statement hides an exponent statement. Always translate between the two forms to build intuition.
2 · Special Bases: Common & Natural Logarithms
Notation Base Name Example
log(x) 10 Common log log(1000) = 3 since 10^3 = 1000
ln(x) e ≈ 2.71828 Natural log ln(e^4 ) = 4
log 2 (x) 2 Binary log log 2 (32) = 5 since 2^5 = 32
3 · Properties of Logarithms (with Exponent Proofs)
Each logarithm property corresponds directly to an exponent law. Understanding why a property
works — through the lens of exponents — makes it unforgettable.
Product Rule logb(M · N) = logb(M) + logb(N)
Let logb(M) = p and logb(N) = q, so M = bp^ and N = bq. Then M·N = bp·bq^ = bp+q^ (exponent addition law). Taking logb of both sides gives logb(M·N) = p
- q. Example: log 2 (4 · 8) = log 2 (32) = 5 = log 2 (4) + log 2 (8) = 2 + 3.
Quotient Rule logb(M / N) = logb(M)^ −^ logb(N)
Using M = bp^ and N = bq, we get M/N = bp/bq^ = bp−q^ (exponent subtraction law). Therefore logb(M/N) = p − q. Example: log 10 (1000/10) = log 10 (100) = 2 = 3 − 1.
Power Rule logb(Mn) = n · logb(M)
With M = bp, we have Mn^ = (bp)n^ = bpn^ (exponent multiplication law). So logb(Mn) = pn = n·logb(M). Example: log 2 (8^2 ) = log 2 (64) = 6 = 2 × 3.
Zero Rule logb(1) = 0
Because b^0 = 1 for any valid base b, it follows that logb(1) = 0. The exponent needed to turn the base into 1 is always zero. Example: log 7 (1) = 0 because 7^0 = 1.
Identity Rule logb(b) = 1
Because b^1 = b for any base, the exponent needed to produce b from base b is 1. Example: log 5 (5) = 1 because 5^1 = 5.
Inverse Rules blogb(x)^ = x and logb(bx) = x
These are the cancellation properties showing that exponentiation and logarithm are exact inverses of each other. Applying one then the other always returns the original value. Examples: 10 log(7)^ = 7 ; ln(e^3 ) = 3.
Identity logb(b) = 1 b^1 = b
Inverse blogbx^ = x Definition of inverse
Reciprocal logb(1/x) = −logb(x) x−^1 exponent rule
Change of Base logb(x) = ln(x)/ln(b) Derived from definition
Remember: every logarithm is secretly an exponent. When in doubt, rewrite logb(x) = y as by^ = x and use exponent laws.
