The invented law, why it feels right, and the plot that kills it.
Write log(x + y) = log x + log y on the board and ask whether it is true.
A good number will accept it, because addition appears on both sides and the pattern looks symmetrical. The two curves are nowhere near each other: at x = 4 and y = 25 they are 2 and 1.462.
They even cross at small y, so the invented version is sometimes larger and sometimes smaller. Nothing about them is related, which is the cleanest possible disproof.
A logarithm is a power. Powers add when you multiply. Therefore logs add when you multiply. Said that way the law is a consequence rather than a fact to memorise, and the non-existence of a sum law is obvious.
The point of the third law is to get an unknown out of an exponent, which is the only reason anyone solves 3ₓ = 50. Lead with that use and the laws have a purpose.
| 1. log 4 + log 25 | log 100 = 2. |
| 2. 2 log x − log y | Option 1, log(x²/y). |
| 3. log(x + y) | C, nothing simpler; there is no such law. |
1 markCombining to a single logarithm before solving.
1 markA coefficient becoming a power inside, so 2 log x is log x².
1 markRejecting a solution that makes any argument non-positive.
| They give | What it means |
|---|---|
| 1.462 (Q1) | Computed log 29, the invented law. Exactly the error the figure targets. |
| 29 (Q1) | Added the numbers instead of multiplying them. |
| Option 2 (Q2) | Made 2 log x into log 2x. The coefficient is a power, not a multiplier. |
| Option 3 (Q2) | Subtracted inside rather than dividing. |
| A (Q3) | The invented law itself. |
"Which base do I use?" In examinations, 10 or e. Both are on the calculator and the laws are identical for either.
"Why is there no sum law?" Because adding two numbers does nothing tidy to their powers. Ask them to find one and they will see why nobody has.
"Can log be negative?" The value can, for an argument between 0 and 1. The argument cannot be zero or negative, which is a different statement and a common confusion.
| What is happening | |
|---|---|
| 1 | Is log(x+y) = log x + log y? Plot it. |
| 2 | The three laws, derived from powers rather than quoted. |
| 3 | Combining and splitting expressions. |
| 4 | Solving exponential equations, which is what the laws are for. |
| 5 | Checking arguments stay positive. |
Do not list the laws without the reason they hold. The invented fourth law comes straight back otherwise.
Do not let a solution through without checking the argument of every log in the original equation.
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.