A logarithm is an exponent. If $y=2^{x}$ turns the power $3$ into the value $8$, the base-2 logarithm turns $8$ back into $3$, because $3$ is the power to which $2$ must be raised to give $8$: $\log_{2}8=3$. This editable lesson builds the subject in that order — inverse first, graphs second, laws third — and proves every law rather than listing it. It includes three drawn figures, ten proved laws, worked examples, fifteen exercises, and a separate teacher key.
The definition, and why it is restricted
Let $a>0$ with $a\ne1$. For $x>0$, $\log_{a}x$ is the unique real $y$ with $a^{y}=x$:
$$ y=\log_{a}x \iff a^{y}=x . $$
Each restriction is forced. We need $a>0$ so that $a^{y}$ is defined for every real $y$; we need $a\ne1$ because $1^{y}=1$ never takes any other value; and we need $x>0$ because $a^{y}>0$ always, so no exponent can produce zero or a negative number.
Reading the equivalence in both directions gives the inverse identities $a^{\log_{a}x}=x$ for $x>0$ and $\log_{a}(a^{y})=y$ for real $y$. Setting $y=0$ and $y=1$ gives $\log_{a}1=0$ and $\log_{a}a=1$ for every admissible base.
Domain, range, and the two shapes
For $a>0$ with $a\ne1$, the map $y\mapsto a^{y}$ is a bijection from $\mathbb{R}$ onto $(0,\infty)$. Its inverse $\log_{a}$ therefore has domain $x>0$ and range all of $\mathbb{R}$, with the $y$-axis as a vertical asymptote and $(1,0)$ on every curve.
Monotonicity is inherited from the exponential. If $a>1$ then $\log_{a}$ is strictly increasing; if $0<a<1$ it is strictly decreasing. The lesson draws $y=\log_{2}x$ and $y=\log_{1/2}x$ on one set of axes so the contrast is visible, and the two are exact mirror images because
$$ \log_{1/a}x=-\log_{a}x . $$
Strict monotonicity is what makes equations solvable: $\log_{a}u=\log_{a}v$ forces $u=v$.
The ten laws
With $a,b,c$ admissible bases, $x,y>0$, and $p,q$ real with $q\ne0$:
| Law | Statement |
|---|---|
| Product | $\log_{a}(xy)=\log_{a}x+\log_{a}y$ |
| Quotient | $\log_{a}(x/y)=\log_{a}x-\log_{a}y$ |
| Power | $\log_{a}(x^{p})=p\log_{a}x$ |
| Power of the base | $\log_{a^{q}}x=\frac{1}{q}\log_{a}x$ |
| Change of base | $\log_{a}x=\log_{c}x,/,\log_{c}a$ |
| Reciprocal | $\log_{a}b=1/\log_{b}a$ |
| Product of two logarithms | $(\log_{a}b)(\log_{b}c)=\log_{a}c$ |
| Chain of logarithms | $(\log_{a_{1}}a_{2})\cdots(\log_{a_{n-1}}a_{n})=\log_{a_{1}}a_{n}$ |
| Base change of a power | $a^{b}=c^{,b\log_{c}a}$ |
| Exchanging base and argument | $a^{\log_{b}c}=c^{\log_{b}a}$ |
Every proof in the lesson follows one pattern: name the logarithms, convert to index form, apply an index law, convert back. Writing $u=\log_{a}x$ and $v=\log_{a}y$, the product law is immediate: $xy=a^{u}a^{v}=a^{u+v}$, so $\log_{a}(xy)=u+v$. Change of base comes from taking $\log_{c}$ of $a^{\log_{a}x}=x$, which gives $(\log_{a}x)(\log_{c}a)=\log_{c}x$; the division is legal precisely because $a\ne1$ makes $\log_{c}a$ nonzero. The last law is the prettiest: writing $a=b^{\log_{b}a}$ gives $a^{\log_{b}c}=b^{(\log_{b}a)(\log_{b}c)}$, whose exponent is symmetric in $a$ and $c$, so the two may be exchanged.
Worked examples
Expanding, $\log_{a}\frac{x^{3}\sqrt{y}}{z^{2}}=3\log_{a}x+\frac12\log_{a}y-2\log_{a}z$.
Evaluating with the power-of-the-base law, $\log_{4}8=\log_{2^{2}}2^{3}=\frac32$, which checks against $4^{3/2}=8$.
Collapsing a product with the chain law, $(\log_{3}2)(\log_{2}9)=\log_{3}9=2$.
Simplifying an exponential, $2^{\log_{4}9}=2^{\frac12\log_{2}9}=2^{\log_{2}3}=3$.
Representative exercises
- Evaluate $\log_{2}32$, $\log_{9}27$ and $\log_{1/3}81$.
- Solve $\log_{3}(x-1)+\log_{3}(x+1)=1$, testing the domain.
- Simplify $(\log_{2}3)(\log_{3}4)(\log_{4}5)(\log_{5}8)$.
- Solve $\log_{2}x+\log_{4}x+\log_{8}x=11$.
- Given $\log_{12}27=a$, express $\log_{6}16$ in terms of $a$.
- Solve $(\log_{x}2)(\log_{2x}2)=\log_{4x}2$.
- Prove that $\log_{2}3$ is irrational.
- Show that $\log_{2}3>\log_{3}5$ without a calculator.
- Evaluate $\frac{1}{\log_{2}100!}+\frac{1}{\log_{3}100!}+\cdots+\frac{1}{\log_{100}100!}$.
- Solve $x^{\log_{10}x}=100x$.
The teacher key solves all fifteen exercises, records the restriction each answer depends on, lists the misconceptions this topic reliably produces, and supplies a compact scoring guide.
Classroom use and adaptation
The lesson is designed for a class that already knows the index laws for real exponents. Sections 1 and 2 are direct instruction, the proofs in section 3 can be split between board work and set reading, and the exercises give individual evidence. A class short of time can prove the first five laws and set the remaining five as guided exercises, since each is a short consequence of the change-of-base law.
Everything is editable: the base used in the opening example, the two bases drawn in the comparison graph, which laws are proved in class, the number of exercises, and the amount of answer space. The student lesson and the teacher key are separate files, so the key can be exported and distributed on its own.
References and teaching notes
The two Common Core standards cited, HSF-LE.A.4 and HSF-BF.B.5, cover solving exponential equations with logarithms and the inverse relationship between exponents and logarithms; OpenStax Algebra and Trigonometry 2e is cited for the standard treatment of logarithmic functions.
The proofs assume the index laws for real exponents and the fact that $x\mapsto a^{x}$ is a strictly monotone bijection onto the positive reals. The lesson stays with real logarithms; complex logarithms, the natural logarithm as an integral, and differentiation are natural follow-on topics.