Logarithm formulas & rules.
Simplify equations using algebraic logarithm properties. Toggle through rules below, or print to generate a clean PDF formula sheet.
Logarithm Formula Sheet Cheat Sheet
Product Property
logb(x • y) = logb(x) + logb(y) Quotient Property
logb(x / y) = logb(x) - logb(y) Power Property
logb(xk) = k • logb(x) Change of Base Property
logb(x) = logc(x) / logc(b) Identity Properties
logb(b) = 1 • logb(1) = 0 Inverse Exponent Cancellation
blogb(x) = x • logb(bx) = x How Exponents and Logarithms Work Together
Logarithms and exponents are inverse functions—they do the opposite of each other. Just like addition and subtraction, or multiplication and division, one "undoes" the other.
by = x
The exponent takes the base and the power and multiplies the base that many times to get the final result.
Example: 23 = 2 × 2 × 2 = 8
logb(x) = y
The logarithm takes the base and the result and tells you what power you had to raise the base to.
Example: log2(8) = 3
Because they are opposites, putting a logarithm inside its matching exponential base (or vice-versa) cancels them out:
Example: log10(105) = 5 and 10log10(100) = 100
The Pattern of Logarithms
Logarithms might seem abstract at first, but positive, zero, and negative logarithms follow a very simple pattern. Here is how it works for Common Logarithms (base 10):
| Type of Log | Number | Multiplication / Division Pattern | Base-10 Logarithm |
|---|---|---|---|
| Positive | 1,000 | 1 × 10 × 10 × 10 (three tens) | log10(1000) = 3 |
| Positive | 100 | 1 × 10 × 10 (two tens) | log10(100) = 2 |
| Positive | 10 | 1 × 10 (one ten) | log10(10) = 1 |
| Zero | 1 | 1 (no tens multiplied/divided) | log10(1) = 0 |
| Negative | 0.1 | 1 ÷ 10 (divide by one ten) | log10(0.1) = -1 |
| Negative | 0.01 | 1 ÷ 10 ÷ 10 (divide by two tens) | log10(0.01) = -2 |
| Negative | 0.001 | 1 ÷ 10 ÷ 10 ÷ 10 (divide by three tens) | log10(0.001) = -3 |
Key Takeaways:
• Positive logs tell you how many times to multiply by the base.
• Zero logs always result from taking the log of 1, because anything to the power of 0 is 1.
• Negative logs tell you how many times to divide by the base (e.g., log5(0.008) = -3 since 1 ÷ 5 ÷ 5 ÷ 5 = 0.008).
What is natural logarithm?
A natural logarithm is a logarithm that has the mathematical constant e as its base. The constant e (Euler's number) is an irrational number approximately equal to 2.718281828.
Instead of writing loge(x), mathematicians and calculators use the dedicated notation ln(x):
Natural logarithms are fundamental in calculus and calculus-based physics because they describe processes of continuous growth and decay.
Key properties of natural logarithms include:
- ln(e) = 1 (Because e1 = e)
- ln(1) = 0 (Because e0 = 1)
- eln(x) = x (Cancellation law)
Derivative of logarithm.
In calculus, the rate of change of a logarithmic function is critical. The derivative laws for logarithms are defined as follows:
For the natural log, the derivative is simply reciprocal.
Derived by rewriting logb(x) as ln(x) / ln(b) using change of base, and differentiating.
Chain Rule Application
When taking the derivative of a function inside a logarithm, apply the chain rule: