The logarithm and exponent rules

log ruleslaws of logarithmsexponent ruleschange of basesolving for an exponent

Product, quotient and power rules for logarithms, change of base, and the exponent identities they mirror — the toolkit for solving for an exponent.

Logarithm of a Product

\log_b(xy) = \log_b x + \log_b y

The logarithm of a product equals the sum of the logarithms, the identity that once turned multiplication into simple addition.

Logarithm of a Quotient

\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y

The logarithm of a quotient equals the difference of the logarithms, turning a division problem into a simple subtraction.

Logarithm of a Power

\log_b\!\left(x^{n}\right) = n \log_b x

The logarithm of a power pulls the exponent out in front, the identity that unlocks any equation with the unknown in an exponent.

Logarithm Change of Base

\log_b x = \frac{\ln x}{\ln b}

Value of a logarithm in any base, rewritten with natural logarithms so an ordinary calculator can evaluate it directly.

Solve an Exponential Equation for the Exponent

x = \frac{\ln (y / a)}{\ln b}

Solves a·bˣ = y for the unknown exponent x using logarithms, the standard move behind every growth and decay problem.

Power of a Power

\left( b^{m} \right)^{n} = b^{mn}

Raising a power to another power multiplies the two exponents, the exponent law behind every nested exponential expression.

nth Root as a Fractional Exponent

y = \sqrt[n]{x} = x^{1/n}

The nth root of a number written as a fractional exponent, so that roots and powers obey exactly the same set of rules.

How they fit together

A logarithm is an exponent, so every log rule is an exponent rule seen from the other side. Multiplying powers adds exponents, so the log of a product is a sum of logs; dividing subtracts, so the log of a quotient is a difference; raising to a power multiplies, so the log of a power brings the exponent down in front. That last one is the workhorse — it is what turns an unknown stuck in an exponent into an ordinary coefficient you can divide by, which is how half-lives, doubling times and compound-interest terms get solved.

Reach for the power rule whenever the unknown is upstairs, and for change of base whenever the base is not 10 or e, because that is all your calculator offers — log₂1000 is just log 1000 / log 2. The mistakes are almost always invented rules. log(a + b) is not log a + log b, and there is no rule for the log of a sum at all; (log a)/(log b) is not log(a − b); and (a + b)ⁿ is not aⁿ + bⁿ. The domain also bites: the argument of a log must be strictly positive, so an algebraically correct manipulation can hand you a root that has to be thrown away.