Exponential and logarithmic functions, grade 12 (4 units)

Two new families of functions: exponential functions and logarithmic functions.Subjects

They describe many processes in nature and in the economy – savings in a bank, a virus spreading, a radioactive substance decaying. The course builds on the differential and integral calculus of grades 10 and 11 (4 units): the derivative of a polynomial, of a root and of a rational function, the tangent, investigating a function, the integral and the area. The topics, in order:

powers: all the laws of powers, a rational exponent included;
the exponential function $\displaystyle a^x$ : its properties, transformations, exponential equations and inequalities;
the logarithm: its definition, the laws of logarithms, the logarithmic function, logarithmic equations and inequalities;
growth and decay: $\displaystyle {f(t)}={f(0)\cdot q^t}$ , a uniform unit of time and the half-life;
the number $\displaystyle e$ : the derivatives of $\displaystyle e^x$ and of $\displaystyle \ln x$ , investigating functions, the graph of a function and the graph of its derivative;
integrals that lead to $\displaystyle e^x$ and to $\displaystyle \ln x$ , and areas.

There are two kinds of lessons:

learning lessons – an explanation, worked examples and a few questions to try;
practice lessons – questions only.

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Last updated on 18.10.2026

Formulas: exponential and logarithmic functions, grade 12 (4 units)

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Powers

A product of powers with the same base
$\displaystyle a^{x}\cdot a^{y}$ $\displaystyle =$ $\displaystyle a^{x+y}$
A quotient of powers with the same base
$\displaystyle \frac{a^{x}}{a^{y}}$ $\displaystyle =$ $\displaystyle a^{x-y}$
A power of a power
$\displaystyle (a^{x})^{y}$ $\displaystyle =$ $\displaystyle a^{x\cdot y}$
A power of a product
$\displaystyle (a\cdot b)^{x}$ $\displaystyle =$ $\displaystyle a^{x}\cdot b^{x}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{x}$ $\displaystyle =$ $\displaystyle \frac{a^{x}}{b^{x}}$
A negative exponent
$\displaystyle a^{-x}$ $\displaystyle =$ $\displaystyle \frac{1}{a^{x}}$
The exponent 0
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$
A rational exponent
$\displaystyle a^{\frac{m}{n}}$ $\displaystyle =$ $\displaystyle \sqrt[n]{a^{m}}$ $\displaystyle a>0$

Logarithms (within the domain)

The definition of the logarithm
$\displaystyle \log_{a}x=b\iff a^{b}=x$ $\displaystyle a>0,\ a\neq1,\ x>0$
$\displaystyle \log_{a}(a^{b})$ $\displaystyle =$ $\displaystyle b$
$\displaystyle a^{\log_{a}x}$ $\displaystyle =$ $\displaystyle x$
The logarithm of a product
$\displaystyle \log_{a}x+\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}(x\cdot y)$
The logarithm of a quotient
$\displaystyle \log_{a}x-\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}\left(\frac{x}{y}\right)$
The logarithm of a power
$\displaystyle \log_{a}(x^{b})$ $\displaystyle =$ $\displaystyle b\cdot\log_{a}x$
The natural logarithm – to the base $\displaystyle e$
$\displaystyle \ln x=\log_{e}x$

Growth and decay

The amount after $\displaystyle t$ units of time
$\displaystyle f(t)=f(0)\cdot q^{t}$
The growth / decay factor – $\displaystyle p$ the percent per unit of time
$\displaystyle q=1\pm\frac{p}{100}$
The half-life $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle \frac{1}{2}$

Derivatives

$\displaystyle (e^{x})^{\prime}$ $\displaystyle =$ $\displaystyle e^{x}$
$\displaystyle (\ln x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x}$
The derivative of $\displaystyle e^{f(x)}$
$\displaystyle \left[e^{f(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle e^{f(x)}\cdot f^{\prime}(x)$
The derivative of $\displaystyle \ln(f(x))$
$\displaystyle \left[\ln(f(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)}{f(x)}$
The derivative of a product
$\displaystyle \left[f(x)\cdot g(x)\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(x)\cdot g(x)+f(x)\cdot g^{\prime}(x)$
The derivative of a quotient
$\displaystyle \left[\frac{f(x)}{g(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)\cdot g(x)-f(x)\cdot g^{\prime}(x)}{\left[g(x)\right]^{2}}$
The derivative of a composite function
$\displaystyle \left[f(u(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(u)\cdot u^{\prime}(x)$
The tangent’s equation at the point where $\displaystyle {x}={a}$
$\displaystyle y-f(a)=f^{\prime}(a)(x-a)$

Integrals and the area

$\displaystyle \int e^{x}\,dx=e^{x}+C$
$\displaystyle \int e^{mx+b}\,dx=\frac{e^{mx+b}}{m}+C$ $\displaystyle m\neq0$
In any interval where $\displaystyle x\neq0$
$\displaystyle \int\frac{1}{x}\,dx=\ln|x|+C$
$\displaystyle \int\frac{1}{mx+b}\,dx=\frac{\ln|mx+b|}{m}+C$ $\displaystyle m\neq0$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$