Exponential and Logarithm Functions

An exponential function is simply something that keeps getting multiplied by the same number, like money earning interest or a drug wearing off by half every few hours. A logarithm is the reverse question: how many times did I have to multiply? The first formula has three pieces: $b$ (the base, the number you keep multiplying by), $x$ (how many times you multiply, the exponent) and $\log_b y$ (the exponent you'd need to reach $y$). So $\log_2 8 = 3$, because three 2's multiplied together make 8. The log rules look like a lot to memorize, but they're just exponent rules seen from the other side.

Think about why $\ln(xy) = \ln x + \ln y$. A log is an exponent, and when you multiply powers of the same base, you add the exponents: $e^a\cdot e^b = e^{a + b}$. So the exponent for $xy$ is the exponent for $x$ plus the exponent for $y$. The other two rules come from the same place: dividing powers subtracts exponents, and raising a power to a power multiplies them.

A common mistake. It's tempting to split $\ln(x + y)$ into $\ln x + \ln y$, because it looks like the product rule. It doesn't work: logs turn products into sums, and there's no rule at all for the log of a sum. For example, $\ln(1 + 1) = \ln 2 \approx 0.69$, but $\ln 1 + \ln 1 = 0$.

The graphs. $e^x$ is always positive, passes through $(0, 1)$, races to infinity on the right and flattens toward 0 on the left. $\ln x$ is its mirror image across $y = x$: defined only for $x > 0$, through $(1, 0)$, slowly rising.

Undoing each other. $e^{\ln x} = x$ for $x > 0$, and $\ln(e^x) = x$ for every $x$.

Change of base. $\log_b x = \dfrac{\ln x}{\ln b}$, which is how calculators handle any base.

Solving equations.

• Exponential: isolate the exponential, then take $\ln$ of both sides.
• Logarithmic: combine into a single log, then exponentiate. Always check answers in the original equation, because logs of negative numbers are undefined.

pH, decibels and the Richter scale are logarithmic, because the quantities they measure span many powers of ten. Radioactive decay, compound interest and early population growth are exponential.