Laws of Logarithms worksheet - Free Printable

Let’s go through each logarithmic rule and match the correct pair from the given options. The goal is to find two rectangles (one blue, one yellow) that correctly demonstrate the stated logarithmic rule.

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1. Base Rule

> This likely refers to: `log_b(b) = 1` — the logarithm of the base itself is always 1.

- Blue box: `log₆24 =`
- Yellow box: `3 * log₃2`

✘ Neither matches the Base Rule.

Wait — let’s look at #5:

5. Power Rule

> `log_b(a^c) = c * log_b(a)`

- Blue: `log₄4 =`
- Yellow: `1`

This is not the Power Rule — this is the Base Rule: `log₄4 = 1`.

So perhaps #5 is mislabeled? Let’s check all carefully.

Actually, let’s re-evaluate each rule with standard logarithmic identities.

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## ✔ Correct Matching:

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1. Base Rule → `log_b(b) = 1`

Look for: `log_something(same something) = 1`

→ #5: `log₄4 = 1` ← This fits the Base Rule!

But it’s labeled “Power Rule” — so maybe the labels are mixed up?

Wait — the worksheet says:

> Find the two rectangles that match the rule below

Each row has a rule name, then two drop zones, and then two boxes (blue and yellow) to drag into them.

We must match the rule name with the correct identity.

Let’s do it properly.

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## 🔍 Step-by-step matching:

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1. Base Rule ⇒ `log_b(b) = 1`

Which pair shows this?

- `log₄4 = 1` → YES! That’s the Base Rule.

But in the worksheet, this is under #5 Power Rule.

So perhaps the worksheet has mismatched labels? Or we need to match based on content, not label?

The instruction says: “Find the two rectangles that match the rule below”

Meaning: For each numbered rule, find which blue/yellow pair demonstrates that rule.

So let’s ignore the position and match based on math.

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Let’s list all blue/yellow pairs and assign them to the correct rule.

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Pair A: `log₆24 = 3 * log₃2`

→ Not a standard rule. `log₆24` ≠ `3 log₃2`. (Try calculating: log₆24 ≈ 1.76, 3 log₃2 ≈ 3*0.63≈1.89 — close but not equal, and not a rule.)

✘ Invalid or misleading.

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Pair B: `log₆24 = 0`

→ False. log₆24 > 0 since 24 > 1.
✘ Incorrect.

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Pair C: `log₉1 = log₆4 + log₆6`

→ Left: `log₉1 = 0` (since log_b(1) = 0)
→ Right: `log₆4 + log₆6 = log₆(4*6) = log₆24` ≠ 0
✘ Not equal → invalid.

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Pair D: `log₈(1/5) = log₁₀7 / log₁₀5`

→ Left: `log₈(1/5) = -log₈5`
→ Right: `log₁₀7 / log₁₀5 = log₅7` (Change of Base)
→ So: `-log₈5` vs `log₅7` — not equal.
✘ Invalid.

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Pair E: `log₄4 = 1`

→ ✔ Base Rule: `log_b(b) = 1` → Perfect match.

So this should be under Base Rule, but it's listed under #5 "Power Rule".

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Pair F: `log₅7 = log₆48 - log₆2`

→ Right side: `log₆(48/2) = log₆24`
→ So: `log₅7 = log₆24`? No, different bases.
✘ Not equal.

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Pair G: `log₃8 = - * log₈5`

→ Probably typo: maybe meant `-log₈5`? But `log₃8` is positive, `-log₈5` is negative → not equal.
✘ Invalid.

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Pair H: `log₅7 = 1 / log₇5`

→ ✔ Reciprocal Rule: `log_b(a) = 1 / log_a(b)`
→ So `log₅7 = 1 / log₇5` → TRUE!

This matches Reciprocal Rule (#6).

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Now, let’s also check:

What about Product Rule? #3

Product Rule: `log_b(xy) = log_b(x) + log_b(y)`

Is there any pair showing this?

Look at Pair C again: `log₉1 = log₆4 + log₆6` — left is 0, right is log₆24 — not equal.

No valid Product Rule shown.

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Quotient Rule? #4

Quotient Rule: `log_b(x/y) = log_b(x) - log_b(y)`

Pair D: `log₈(1/5) = ...` — left is quotient, but right is change of base — doesn’t match.

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Power Rule? #5

Power Rule: `log_b(a^c) = c * log_b(a)`

None of the pairs show this directly.

Except... maybe Pair A: `log₆24 = 3 * log₃2` — but as we saw, not true.

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Change of Base Rule? #7 and #8

Change of Base: `log_b(a) = log_c(a) / log_c(b)`

Pair D: `log₈(1/5) = log₁₀7 / log₁₀5` — no, that’s `log₅7`, not related.

Pair H: `log₅7 = 1 / log₇5` — that’s Reciprocal, not Change of Base.

Wait — what if we consider:

✔ Valid Matches:

Only two pairs are mathematically correct:

1. `log₄4 = 1` → Base Rule
2. `log₅7 = 1 / log₇5` → Reciprocal Rule

All others are either false or don't match the named rule.

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## 🎯 Final Answer:

The two correct rectangles that match their respective rules are:

- For Base Rule (#1):
→ Blue: `log₄4 =`
→ Yellow: `1`

- For Reciprocal Rule (#6):
→ Blue: `log₅7 =`
→ Yellow: `1 / log₇5`

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✔ These are the only two pairs that are both mathematically correct and match the stated logarithmic rule.

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## 📝 Note:
The worksheet seems to have some mismatches in labeling (e.g., Base Rule is shown under #5 labeled “Power Rule”), but based on mathematical correctness and matching the rule name, these are the only valid pairs.

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## ✔ Final Boxed Answer:

The two correct matches are:

- Base Rule: `log₄4 = 1`
- Reciprocal Rule: `log₅7 = \frac{1}{\log_7 5}`

These are the only two rectangles that correctly demonstrate their respective logarithmic rules.