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If someone tells you 2x=322^x = 32, your brain quickly figures out that x=5x = 5 by multiplying 2 repeatedly.
An inverse operation is an operation that reverses the effect of another, just like subtraction undoes addition and division undoes multiplication. But how do we algebraically isolate an exponent when it's stuck up in the power?
To undo an exponent and solve for the power directly, mathematicians invented a dedicated inverse function.

The Logarithmic Question

In 1614, Scottish mathematician John Napier introduced logarithms to simplify tedious astronomical calculations. The conceptual meaning of log⁑b(x)\log_b(x) asks: "To what power must base bb be raised to produce the value xx?"
By the fundamental definition, for any positive base bb (b>0,bβ‰ 1b > 0, b \neq 1) and positive number x>0x > 0, the equation log⁑b(x)=y\log_b(x) = y is equivalent to by=xb^y = x.
Through this component mapping, the base bb remains the base, the exponent yy becomes the isolated output, and the power xx becomes the input argument.
What happens when our exponents are negative or fractional? Let's see how the exact same rule applies.

Converting Exponentials to Logarithms

To convert by=xb^y = x into log⁑b(x)=y\log_b(x) = y, identify the base bb, place it in the subscript of log⁑\log, bring the result xx inside as the argument, and set the whole expression equal to the exponent yy.
Let's convert 5βˆ’2=1255^{-2} = \frac{1}{25}. The base is 55, the exponent is βˆ’2-2, and the result is 125\frac{1}{25}. Writing this in logarithmic form gives log⁑5(125)=βˆ’2\log_5\left(\frac{1}{25}\right) = -2.