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"For the next generation"
© 2025 Astra.si. All rights reserved.
"For the next generation"
A logarithm is the inverse mathematical operation of exponentiation. If we have a power of the form
a^b = c,
then the logarithm of the number c with base a is equal to b, which we write as:
logₐ(c) = b.
This means: the logarithm answers the question, "To what power must we raise the base a to obtain the number c?"
For example:
log₂(8) = 3, because 2^3 = 8.
base (a): the number being raised to a power (a > 0, a ≠ 1),
argument (c): the number whose logarithm we are finding (c > 0),
result (b): the exponent that shows how many times the base is used in multiplication.
Common (base 10) logarithm: base is 10 → log(c) = log₁₀(c)
Natural logarithm: base is e (≈ 2.718) → ln(c) = logₑ(c)
Logarithmic expressions follow specific calculation rules:
logₐ(x*y) = logₐ(x) + logₐ(y)
logₐ(x/y) = logₐ(x) − logₐ(y)
logₐ(x^n) = n * logₐ(x)
logₐ(a) = 1
logₐ(1) = 0
A logarithm is not defined for negative arguments or for a base equal to 1, because in those cases the exponent cannot be determined in the usual way.
Find:
log₃(81)
Since 3^4 = 81, we get:
log₃(81) = 4
Another example:
log₁₀(1000) = 3, since 10^3 = 1000.
A logarithm is an essential mathematical concept that links powers, exponents, and multiplication in an inverse way. Thanks to its properties, it makes it possible to simplify calculations with very large or very small numbers and to break down exponential expressions into linear components. Understanding logarithms means understanding the relationship between growth, powers, and the inverse of exponentiation.
