Why must a logarithm's input be positive?

Think of a logarithm as asking an exponent question: logₐx asks, “What real number y makes aʸ = x?” For a real logarithm, the base must be positive and cannot be 1, and its input x must be positive. [S1]

The input restriction follows from the exponential equation. If a is positive, then aʸ is positive for every real y. So there is no real exponent that makes aʸ equal zero or a negative number. The logarithm is not rejecting those inputs arbitrarily; there is no real-number answer to the exponent question. [S1]

Use base 2 and rewrite each logarithm as an exponential equation:

1. Valid: log₂8 = 3 because 2³ = 8. The input, 8, is positive, and the exponent question has a real answer. [S1]
2. Invalid over the real numbers: log₂0 would ask for a real y with 2ʸ = 0. No real power of 2 equals zero, so the logarithm has no real value. [S1]
3. Also invalid over the real numbers: log₂(−8) would ask for a real y with 2ʸ = −8. A positive base raised to a real power is positive, not negative, so there is no real answer. [S1]

The phrase “over the real numbers” matters: these examples are about real logarithms, not every possible mathematical system. The cited definition establishes the restriction in the real setting; it does not establish rules for other settings. [S1]

First check the input, also called the argument: it must be greater than zero. Then check that the base is positive and not equal to 1, as required for the real logarithm described here. [S1] If the input is zero or negative, stop: there is no real logarithm to evaluate. If it is positive and the base meets those conditions, rewrite logₐx = y as aʸ = x to identify the real exponent.

For example, with log₃(−9), the input is −9, so it fails the positive-input check immediately. With log₃9, the input passes, and rewriting gives 3ʸ = 9; since 3² = 9, the value is 2. This check tells you whether a real logarithm is permitted; it does not by itself calculate every logarithm.

Use this copyable explanation:

“[Expression] is a real logarithm with base [a] and input [x]. The base [is/is not] positive and not equal to 1; the input [is/is not] greater than zero. Since a real logarithm requires all those conditions, [the expression is allowed / the expression has no real value]. To check an allowed example, rewrite it as [a]ʸ = [x] and solve for y.”

For instance: “log₂8 is a real logarithm with base 2 and input 8. The base meets the conditions, and the input is greater than zero, so it is allowed. Rewriting it as 2ʸ = 8 gives y = 3.” For a disallowed example: “log₂0 has input 0, which is not greater than zero, so it has no real value.” Keep the conclusion limited to real logarithms. [S1]