What a logarithm is, before any laws, and why the graph does the explaining.
Say the sentence before you show anything: 2 to the WHAT gives 32?
That sentence is the definition. Everything else in logarithms is consequence, and students who have it never need to be told what a log is again.
Then slide to x = 5 and point at the two dots: the same pair of numbers, (5, 32) and (32, 5). Inverse stops being a word and becomes a reflection they can see.
3 × 2t = 96 is one line if you divide first and three lines of mess if you do not. This is the single most useful procedural habit in the sub-topic.
And estimate before accepting. log₁₀50 must be between 1 and 2 because 50 is between 10 and 100. That catches a mistyped calculator entry instantly and costs nothing.
| 1. log₁₀1000 | 3. |
| 2. 3 × 2t = 96 | 2t = 32, so t = 5. |
| 3. log₁₀50 | B, between 1 and 2. It is 1.699. |
1 markIsolating the power before any log is taken.
1 markNegative powers written as reciprocals, never as negative answers.
1 markA log answer of a sensible size, estimated before it is accepted.
| They give | What it means |
|---|---|
| 1000 (Q1) | Gave the number back. They have not yet got that a log IS the power. |
| 2.9957 (Q1) | Used ln. Worth distinguishing: the method is right and the base is wrong. |
| 6.585 (Q2) | Took logs with the 3 still attached. |
| 32 (Q2) | Stopped one line early, at 2ᵗ = 32. |
| C or D (Q3) | Thinks a log is near the size of the number. The estimate habit fixes this faster than any explanation. |
"What is e?" A number, about 2.718, that turns up wherever growth compounds continuously. At this stage they need to know ln means log base e and nothing more.
"Can I take the log of a negative number?" No. No power of 10 produces a negative, so it does not exist. If one appears, something earlier is wrong, and that is useful information.
"Why do calculators only have log and ln?" Because those two cover everything via the change of base rule, which they meet later. For now, 2ˣ questions are solved by reading, not by a button.
| What is happening | |
|---|---|
| 1 | The sentence, then the two dots on the graph. |
| 2 | Laws of exponents, which most will half remember. |
| 3 | Logs as the inverse, with the aₓ = b equivalence written both ways. |
| 4 | Solving exponential equations, isolating the power every time. |
| 5 | Estimating logs before computing them. |
Do not start with the laws of logarithms. They are meaningless until the definition is secure, and they arrive properly in a later topic anyway.
Do not let "log" be a button. Every log question at this level can be read aloud as a question about a power.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.