Formulas and properties of logarithms: complete cheat sheet for the EGE and OGE

Logarithm is the exponent to which the base must be raised to get a given number: log_ab = c if a^c = b. Basic properties: log(x·y) = log x + log y, log(x/y) = log x − log y, log xⁿ = n·log x. Types: natural (ln, base e ≈ 2.718) and decimal (lg, base 10).

Formulas and properties of logarithms — a basic cheat sheet for schoolchildren and students. Here you will find: the definition of a logarithm, the main properties (logarithm of a product, quotient, power, root), the change of base formula, the derivative of a logarithm, as well as types of logarithms — natural (ln) and decimal (lg). The material is suitable for preparing for the EGE and OGE, as well as for first-year students.

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About the author

The material was prepared by Alexey Pavlov, a mathematics teacher. The formulas and properties of logarithms have been tested in practice while preparing students for the EGE and OGE.

On this page:

Definition of logarithm
Types of logarithms
Formulas and properties
Graph of logarithm
Frequently asked questions
Download cheat sheet

Definition of logarithm

Logarithm is a mathematical function inverse to exponentiation. The logarithm of a number b to base a (log_a b) is the exponent to which a must be raised to obtain b.

Conditions: b > 0, a > 0, a ≠ 1.

How to solve a logarithm in practice:

To find the value of log₂8, answer the question: "To what power must 2 be raised to get 8?" Answer: 2³ = 8, so log₂8 = 3.

Similarly: log₃81 = 4, because 3⁴ = 81.

Types of logarithms

Logarithm of a positive number b to base a (log_a b) is the exponent to which a must be raised to obtain b.

Examples of logarithms:

${log}_{2} 8 = 3$ (logarithm of $8$ to base 2), because $2^{3} = 8$

${log}_{7} 49 = 2$ (logarithm of $49$ to base 7), because $7^{2} = 49$

${log}_{5} \frac{1}{5} = -1$ (logarithm of $\frac{1}{5}$ to base 5), because $5^{-1} = \frac{1}{5}$

Decimal logarithm

Decimal logarithm — logarithm to base 10.

$lg b$ — Decimal logarithm

Examples of decimal logarithms:

$lg 100 = 2$ — decimal logarithm of one hundred, because 10² = 100.

$lg 1000 = 3$ , because 10³ = 1000.

👉 Table of decimal logarithms.

Natural logarithm

$ln b$ — natural logarithm (logarithm to base e, $a = e$ ), where e ≈ 2.718.

Example: ln(e²) = 2, because e² = e².

Important: The natural logarithm is often used in higher mathematics, physics, and economics.

👉 Table of natural logarithms.

Formulas and properties of logarithms

For any $a > 0$ , $a \neq 1$ and $b > 0$ , $x > 0$ , $y > 0$ the following properties of logarithms hold.

Basic logarithmic identity

A base a raised to the power of a logarithm with base a equals b.

Example: Example: 2^log₂8 = 2³ = 8.

a^{{log}_{a} b} = b

Logarithm of one

Logarithmic zero. Whatever the base of the logarithm, if the argument is 1, the logarithm is always 0.

Example: Example: log₅1 = 0, because 5⁰ = 1.

{log}_{a} 1 = 0

Logarithm of a number equal to the base

Logarithmic one. If the argument and the base of the logarithm are the same, the value of the logarithm is one.

Example: Example: log₇7 = 1, because 7¹ = 7.

{log}_{a} a = 1

Logarithm of the reciprocal of the base

If the argument of the logarithm is the reciprocal of the base, the value of the logarithm is -1.

Example: Example: log₃(1/3) = -1, because 3⁻¹ = 1/3.

{log}_{a} \frac{1}{a} = - 1

Logarithm of the product of two positive numbers

Sum of logarithms. When multiplying numbers under the logarithm, you can turn them into a sum of two logarithms with the same base.

Example: Example: log₂(8·4) = log₂8 + log₂4 = 3 + 2 = 5.

{log}_{a} (x \cdot y) = {log}_{a} x + {log}_{a} y

Logarithm of a quotient

Logarithm of a quotient. When dividing numbers, we get the difference of two logarithms with the same base.

Example: Example: log₂(8/4) = log₂8 - log₂4 = 3 - 2 = 1.

{log}_{a} \frac{x}{y} = {log}_{a} x - {log}_{a} y

{log}_{a} \frac{1}{y} = {- log}_{a} y

Logarithm of a power of a positive number

Logarithm of a power of a positive number equals the product of the exponent and the logarithm of that number.

Example: Example: log₂8³ = 3·log₂8 = 3·3 = 9.

{log}_{a} x^{n} = n {log}_{a} x

Logarithm of a root of a number

Logarithm of a root equals the quotient of the logarithm of the radicand divided by the index of the root.

Example: Example: log₅√25 = log₅25 / 2 = 2/2 = 1.

{log}_{a} \sqrt[n]{x} = \frac{{log}_{a} x}{n}

Base of the logarithm raised to a power

Example: Example: log₄64 = log₂64 / 2 = 6/2 = 3.

{log}_{a^{n}} x = \frac{{log}_{a} x}{n}

, при

n \neq 0

Example: Example: log₂8 = log₄8² = log₄64 = 3.

{log}_{a} x = {log}_{a^{c}} x^{c}

Change of base formula

Allows you to switch from a logarithm with one base to a logarithm with another base.

Example: Example: log₂8 = log₁₀8 / log₁₀2 ≈ 0.903 / 0.301 = 3.

{log}_{a} x = \frac{{log}_{b} x}{{log}_{b} a}

{log}_{a} x = \frac{1}{{log}_{x} a}

Derivative of a logarithm

Derivative of a logarithmic function with respect to its base equals one divided by the product of the sublogarithmic function and the natural logarithm of the base.

Example: Example: (log₂x)' = 1/(x·ln2).

({log}_{a}, x)' = \frac{1}{x ln a}

Graph of logarithms

By moving the slider, you change the base of the logarithm, which affects the shape of the graph. With a = e (≈ 2.718) you get the natural logarithm, with a = 10 — the decimal one.

y = log₂ x

Frequently asked questions about logarithms

What is a logarithm in simple words?

A logarithm is the exponent to which the base must be raised to get a given number. For example, log₂8 = 3, because 2³ = 8. Simply put, it is the answer to the question: "To what power must the number a be raised to get b?"

What is the difference between decimal and natural logarithm?

The decimal logarithm has base 10 (denoted lg), and the natural logarithm has base e ≈ 2.718 (denoted ln). The decimal logarithm is often used in engineering calculations, the natural one — in higher mathematics, physics, and economics.

What basic properties of logarithms do you need to know for the EGE?

Basic properties: the logarithm of a product equals the sum of logarithms, the logarithm of a quotient — the difference of logarithms, the logarithm of a power — the product of the exponent and the logarithm. It is also important to remember that logₐa = 1 and logₐ1 = 0. All these properties are presented in the table above.

How do you change from one base of a logarithm to another?

To change to a new base, use the formula: logₐx = logᵦx / logᵦa. For example, log₂8 = log₁₀8 / log₁₀2 ≈ 0.903 / 0.301 = 3. This is useful when you need to calculate a logarithm on a calculator that only supports decimal logarithms.

What is the logarithm of 1?

The logarithm of 1 to any base is 0: logₐ1 = 0. This follows from the definition of a logarithm: any number to the zero power is 1.

What is the logarithm base 2?

The logarithm base 2 (log₂) is the exponent to which 2 must be raised to get a given number. For example, log₂8 = 3, log₂16 = 4, log₂32 = 5. For quickly finding values, it is convenient to use a trigonometric or engineering calculator, as well as a table of logarithms.

Can you calculate the logarithm of a negative number?

No, the logarithm of a negative number and the logarithm of zero are not defined in the domain of real numbers. This is because the base to any power gives a positive number, so the argument of the logarithm must always be greater than zero (b > 0).

How do you solve logarithms?

To solve a logarithm, ask the question: "To what power must the base a be raised to get the argument b?" For example, log₃81 = 4, because 3⁴ = 81. For complex cases, use the properties of logarithms: the logarithm of a product, quotient, power, and the change of base formula.

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