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If you type log⁑10(βˆ’5)\log_{10}(-5) into any calculator, you will immediately get a DOMAIN ERROR.
Why does this happen, and why does algebraic manipulation sometimes hand us solutions that math simply forbids?

The Core Constraint

In the expression log⁑b(x)\log_b(x), the input value xx is called the argument, which is the quantity you are taking the logarithm of.
By definition, log⁑b(x)=y\log_b(x) = y means by=xb^y = x. When the base bb is positive (b>0b > 0 and bβ‰ 1b \neq 1), raising it to any real power yy always yields a strictly positive output: by>0b^y > 0.
In 1614, Scottish mathematician John Napier invented logarithms to turn tedious multiplication into simple addition for astronomers, defining them fundamentally through continuous geometric ratios.
Because no real exponent can turn a positive base into zero or a negative number, the argument of any real logarithm must always be strictly positive: argument>0\text{argument} > 0.
What happens when an equation contains multiple logarithmic terms with different algebraic expressions?

Establishing Domain Restrictions

Before you combine logarithms or apply power rules, you must establish the domain restriction by requiring every individual logarithmic argument to be greater than zero.
Consider the equation log⁑2(xβˆ’3)+log⁑2(xβˆ’1)=3\log_2(x - 3) + \log_2(x - 1) = 3. Each argument creates its own inequality condition.