AP Calculus BC

Logarithmic Rules and Functions

Logarithm rules

A logarithm answers "what exponent gives this result." $\log_b N = x$ means $b^x = N$.

Three rules let you combine or split logs, as long as every log in the equation has the same base.

• Product rule: $\log_b(MN) = \log_b M + \log_b N$.
• Quotient rule: $\log_b\dfrac{M}{N} = \log_b M - \log_b N$.
• Power rule: $\log_b M^p = p \log_b M$.

The power rule works in both directions. A coefficient in front of a log can move inside as an exponent, and an exponent inside a log can move out front as a coefficient. That is the move behind most "find $A$" problems: rewrite every term so it has the same log base and argument, then compare coefficients or exponents.

$\ln x$ is $\log_e x$, and $\ln e^n = n$ for the same reason $\log_b b^n = n$: raising $b$ to the $n$ already gives $n$ back.

Logarithmic and exponential functions

A logarithmic function is the inverse of an exponential function: if $f(x) = b^x$, then $f^{-1}(x) = \log_b x$. That inverse relationship explains most of their features.

• $y = \log_b(x - h) + k$ has a vertical asymptote where $x - h = 0$, and it passes through $(h + 1, k)$, since $\log_b 1 = 0$.
• A logarithmic function ($b > 1$) is always increasing, but concave down: it climbs fast at first, then slows down as $x$ grows.
• Since $f$ and $g = f^{-1}$ swap inputs and outputs, if one changes additively while the other changes multiplicatively, the same is true of the inverse function with the roles reversed.

Exponential growth and decay problems ask you to solve for time, which turns into a log once you isolate the exponential.