Mathematics
Logarithm and exponential calculator
Logarithm in any base, exponential equations, antilogarithm and change of base: four readings of the same relation between a power and its exponent.
A logarithm is an exponent
Writing log₂(32) = 5 and writing 2⁵ = 32 say the same thing. A logarithm is not a new operation: it is the inverse question to raising to a power, just as subtraction is the inverse of addition. Keeping that equivalence in mind solves half the exercises on its own, because it lets you move between the logarithmic and the exponential form at any moment.
The existence conditions follow straight from that. The argument must be positive because a power of a positive base never gives zero or a negative number. The base must be positive and different from one: with base one every power would be one, so the question “to what exponent” would have no unique answer. These are not arbitrary restrictions to memorise but consequences of how powers work.
The change of base is the formula that makes everything computable: log_b(x) = ln(x)/ln(b). It is needed because calculators and programming languages implement only the natural and common logarithms, and every other base comes from a division. It is also why the graphs of logarithms in different bases all have the same shape and differ only by a vertical scaling.
Common mistakes
- Treating the logarithm of a sum as the sum of the logarithms: log(a + b) is not log a + log b. The property holds for products: log(a·b) = log a + log b.
- Forgetting the existence conditions in logarithmic equations: the solutions found must always be checked, because one of them can make the argument negative and has to be discarded.
- Confusing the notations: in many textbooks log without a subscript means base 10, while in higher mathematics and in programming languages it means base e. Where the context is unclear, write the subscript.
Frequently asked questions
What is a logarithm?
The exponent the base must be raised to in order to get the argument. log₂(32) = 5 because 2 to the power 5 is 32. Logarithm and power are the same relation read from opposite sides.
How do you solve an exponential equation such as 3ˣ = 81?
Take the logarithm of both sides: x = log₃(81) = 4. When the argument is an exact power of the base you can also simply recognise that 81 = 3⁴.
What is the change-of-base formula?
log_b(x) = log_k(x) / log_k(b), for any new base k that is positive and different from 1. In practice k = e or k = 10, because those are the bases calculators know.
Why must the argument of a logarithm be positive?
Because a power with a positive base is always positive: no exponent makes 2 to the x equal zero or a negative number. That is why the domain of the logarithm is the positive reals.
How this calculation works
Definition: log_b(x) = y is equivalent to bʸ = x, with b > 0, b ≠ 1 and x > 0. Change of base: log_b(x) = ln(x)/ln(b) = log₁₀(x)/log₁₀(b). Properties: log(a·b) = log a + log b; log(a/b) = log a − log b; log(aⁿ) = n·log a; log_b(b) = 1; log_b(1) = 0. Exponential equation bˣ = y: x = log_b(y) = ln y / ln b. Antilogarithm: given log_b(x) = y, x = bʸ. Recurring bases: e ≈ 2.71828 for the natural logarithm ln, 10 for the common logarithm, 2 for the binary logarithm used in computing.
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