Properties of logarithms practice worksheet.

Let’s solve each logarithm expression step by step using the properties of logarithms.

We’ll use these key rules:

1. Product Rule:
`log(a × b) = log a + log b`

2. Quotient Rule:
`log(a / b) = log a - log b`

3. Power Rule:
`log(a^n) = n × log a`

4. Log of 1:
`log(1) = 0` (for any base > 0, ≠ 1)

5. Log of base itself:
`log_b(b) = 1`

---

Now let’s go one by one.

---

1. log(7/3)
→ Use quotient rule:
= log 7 - log 3

---

2. log(9 · x²)
→ First, 9 is 3², so we can write as log(3² · x²)
But even without that, use product rule first:
= log 9 + log(x²)
Then power rule on both:
= 2 log 3 + 2 log x
*(Alternatively, factor out 2: 2(log 3 + log x), but expanded form is fine)*
Final: 2 log 3 + 2 log x

Wait — actually, let’s check if they want it fully expanded. Yes, “expand each logarithm” means break into sum/difference of logs with no exponents or products inside.

So:
log(9 · x²) = log 9 + log(x²) = log(3²) + 2 log x = 2 log 3 + 2 log x

---

3. log(8 · t³)
8 = 2³ → log(2³ · t³) = log(2³) + log(t³) = 3 log 2 + 3 log t
Final: 3 log 2 + 3 log t

---

4. log(√x)
√x = x^(1/2) → use power rule:
= (1/2) log x

---

5. log(6 · z⁴)
= log 6 + log(z⁴) = log(2·3) + 4 log z = log 2 + log 3 + 4 log z
Final: log 2 + log 3 + 4 log z

---

6. log(6 · c⁷)
Same as above:
= log 6 + log(c⁷) = log(2·3) + 7 log c = log 2 + log 3 + 7 log c

---

7. log(9 · x²) ← Wait, this is same as #2? Let me check original image again... Actually in your list, #7 is written as “log(9 · x²)” — yes, duplicate? But maybe typo? Or perhaps different variable? No, looks same.

Wait — looking back at user input:

Actually, in the original problem set, #2 and #7 are both “log(9 · x²)”? That seems odd. Maybe it's a copy-paste error? But since you listed them separately, I’ll assume they’re meant to be solved independently — answer will be same.

But let’s double-check the numbering from your text:

You wrote:

> 1. log(7/3)
> 2. log(9 · x²)
> 3. log(8 · t³)
> 4. log(√x)
> 5. log(6 · z⁴)
> 6. log(6 · c⁷)
> 7. log(9 · x²) ← same as #2
> 8. log(9/x²)
> 9. log(7/t)
> 10. log(9/(2x))
> 11. log(√x) ← same as #4
> 12. log(√x) ← again same?
> 13. log(6 · z⁴) ← same as #5
> 14. log(c⁷)
> 15. log(6 · c⁷) ← same as #6
> 16. log(6c⁷) ← probably same as #15?

Hmm — there are duplicates. But since the instruction says “expand each”, I’ll treat each line as separate and give answers accordingly — even if repeated.

But to avoid confusion, I’ll solve all 16 as listed, noting repeats where applicable.

Let’s continue carefully.

---

8. log(9/x²)
= log 9 - log(x²) = 2 log 3 - 2 log x
Final: 2 log 3 - 2 log x

---

9. log(7/t)
= log 7 - log t
Final: log 7 - log t

---

10. log(9/(2x))
= log 9 - log(2x) = log 9 - [log 2 + log x] = 2 log 3 - log 2 - log x
Final: 2 log 3 - log 2 - log x

---

11. log(√x) ← same as #4
= (1/2) log x

---

12. log(√x) ← again same
= (1/2) log x

---

13. log(6 · z⁴) ← same as #5
= log 6 + log(z⁴) = log(2·3) + 4 log z = log 2 + log 3 + 4 log z

---

14. log(c)
Just power rule:
= 7 log c

---

15. log(6 · c⁷) ← same as #6
= log 6 + log(c⁷) = log(2·3) + 7 log c = log 2 + log 3 + 7 log c

---

16. log(6c⁷) ← assuming this is same as #15 (no dot, but multiplication implied)
= same as above: log 2 + log 3 + 7 log c

---

Now, let’s compile final answers for each numbered item as given.

Note: Some are duplicates — but since the worksheet lists them separately, we provide answers for each.

──────────────────────────────────────

Final Answer:

1. log 7 - log 3
2. 2 log 3 + 2 log x
3. 3 log 2 + 3 log t
4. (1/2) log x
5. log 2 + log 3 + 4 log z
6. log 2 + log 3 + 7 log c
7. 2 log 3 + 2 log x
8. 2 log 3 - 2 log x
9. log 7 - log t
10. 2 log 3 - log 2 - log x
11. (1/2) log x
12. (1/2) log x
13. log 2 + log 3 + 4 log z
14. 7 log c
15. log 2 + log 3 + 7 log c
16. log 2 + log 3 + 7 log c