1.4 · Exponential and logarithmic functions and models

An exponential has a constant multiplicative ratio

For $\displaystyle P(t)=P_0b^t$, a unit increase in time multiplies the amount by $\displaystyle b$. If $\displaystyle b>1$ there is growth; if $\displaystyle 0<b<1$ there is decay. The base cannot be zero or negative for a real exponential defined on all real inputs.

The continuous model $\displaystyle P_0e^{kt}$ uses a continuous relative rate $\displaystyle k$. A discrete increase of $\displaystyle r$ per period uses $\displaystyle b=1+r$; the corresponding continuous coefficient is $\displaystyle \ln(1+r)$, not necessarily $\displaystyle r$.

A logarithm asks for an exponent

The statement $\displaystyle \log_bx=y$ is equivalent to $\displaystyle b^y=x$, for $\displaystyle b>0$, $\displaystyle b\ne1$, and $\displaystyle x>0$. Exponential and logarithmic graphs reflect across $\displaystyle y=x$ when their bases match.

The product, quotient, and power laws follow from exponent rules. They do not turn a logarithm of a sum into a sum of logarithms. In an equation, check every original log argument for positivity.

$$\log_b(uv)=\log_bu+\log_bv$$

Solve models with logarithms

For doubling time $\displaystyle T$, the continuous coefficient satisfies $\displaystyle e^{kT}=2$, so $\displaystyle k=\ln2/T$. For half-life $\displaystyle T$, use $\displaystyle k=-\ln2/T$. Retain exact logarithms until a decimal is required.

A logarithmic domain may remove an algebraic candidate. For $\displaystyle \log_2x+\log_2(x-2)=3$, require $\displaystyle x>2$ before solving $\displaystyle x(x-2)=8$.