Exponentials & Logarithms

Exponential functions

An exponential function has the form $f(x) = a \cdot b^x$ where $b > 0$ and $b \neq 1$.

• If $b > 1$, $f$ grows fast (compound interest, population growth).
• If $0 < b < 1$, $f$ decays toward zero (radioactive decay, cooling).

The most useful base is Euler's number $e \approx 2.71828$. The function $f(x) = e^x$ is the unique exponential whose slope equals its value at every point.

What is a logarithm?

A logarithm is the inverse of an exponential. $\log_b(y) = x$ asks: "to what power do I raise $b$ to get $y$?"

$\log_b(y) = x \iff b^x = y$

Examples.

• $\log_2(8) = 3$ because $2^3 = 8$.
• $\log_{10}(1000) = 3$ because $10^3 = 1000$.
• $\ln(e^5) = 5$ — $\ln$ is shorthand for $\log_e$.

Log rules (the three you must know)

$\log(xy) = \log(x) + \log(y)$ $\log\!\left(\frac{x}{y}\right) = \log(x) - \log(y)$ $\log(x^k) = k \log(x)$

Each rule turns a harder operation (multiply, divide, power) into an easier one (add, subtract, multiply). Pre-calculator, this is how engineers did long multiplications.

Solving exponential equations

To solve for an exponent, take a log of both sides.

Example. $3^x = 81 \implies x = \log_3 81 = 4$.

Example with $e$. $e^{2x} = 7 \implies 2x = \ln 7 \implies x = \frac{\ln 7}{2} \approx 0.973$.

Solving log equations

Exponentiate to undo the log.

Example. $\ln(x) = 3 \implies x = e^3 \approx 20.09$.

Example. $\log_2(x + 1) = 5 \implies x + 1 = 2^5 = 32 \implies x = 31$.

Key takeaways

• $b^x = y \iff \log_b(y) = x$. Logs and exponentials are inverse operations.
• $\ln = \log_e$, the natural log.
• The log rules turn products into sums, quotients into differences, powers into multiplication.
• To isolate an unknown exponent, take a log of both sides.