Printable math worksheet for solving logarithmic equations.

Let's solve each logarithmic equation step by step. The key property we'll use is:

> If $\log_b(A) = \log_b(B)$, then $A = B$, provided $A > 0$ and $B > 0$ (since logs are only defined for positive arguments).

We'll go through each problem one at a time.

---

1) $\log_3(x+7) = \log_3(2x+3)$

Since the bases are the same:
$$
x + 7 = 2x + 3
$$
Solve:
$$
7 - 3 = 2x - x \Rightarrow 4 = x
$$
Check: $x = 4$ → $x+7 = 11 > 0$, $2x+3 = 11 > 0$ ✔
Answer: $x = 4$

---

2) $\log_2(x+20) = \log_2 2$

Same base:
$$
x + 20 = 2 \Rightarrow x = 2 - 20 = -18
$$
Check: $x + 20 = 2 > 0$ ✔, but $x = -18$, so check if input is valid: yes, $x+20 = 2 > 0$.
Answer: $x = -18$

---

3) $\log_4(3x - 2) = \log_4(x + 18)$

Same base:
$$
3x - 2 = x + 18 \Rightarrow 3x - x = 18 + 2 \Rightarrow 2x = 20 \Rightarrow x = 10
$$
Check: $3(10) - 2 = 28 > 0$, $10 + 18 = 28 > 0$ ✔
Answer: $x = 10$

---

4) $\log_5(5x + 6) = 2\log_5 6$

Use log property: $2\log_5 6 = \log_5(6^2) = \log_5(36)$

So:
$$
\log_5(5x + 6) = \log_5(36) \Rightarrow 5x + 6 = 36
\Rightarrow 5x = 30 \Rightarrow x = 6
$$
Check: $5(6)+6 = 36 > 0$ ✔
Answer: $x = 6$

---

5) $\log_4(2x - 1) = \log_4 27$

Same base:
$$
2x - 1 = 27 \Rightarrow 2x = 28 \Rightarrow x = 14
$$
Check: $2(14) - 1 = 27 > 0$ ✔
Answer: $x = 14$

---

6) $\log_3 24 = \log_3(x + 2)$

Same base:
$$
24 = x + 2 \Rightarrow x = 22
$$
Check: $x + 2 = 24 > 0$ ✔
Answer: $x = 22$

---

7) $\log_6(2x + 1) = \log_6(x + 20)$

Same base:
$$
2x + 1 = x + 20 \Rightarrow 2x - x = 20 - 1 \Rightarrow x = 19
$$
Check: $2(19)+1 = 39 > 0$, $19+20 = 39 > 0$ ✔
Answer: $x = 19$

---

8) $\log_4(5x - 3) = \log_4(2x + 36)$

Same base:
$$
5x - 3 = 2x + 36 \Rightarrow 5x - 2x = 36 + 3 \Rightarrow 3x = 39 \Rightarrow x = 13
$$
Check: $5(13)-3 = 62 > 0$, $2(13)+36 = 62 > 0$ ✔
Answer: $x = 13$

---

9) $3\log_3 4 = \log_3(x + 10)$

Use power rule: $3\log_3 4 = \log_3(4^3) = \log_3(64)$

So:
$$
\log_3(64) = \log_3(x + 10) \Rightarrow 64 = x + 10 \Rightarrow x = 54
$$
Check: $x + 10 = 64 > 0$ ✔
Answer: $x = 54$

---

10) $\log_5 5x = \log_5 100$

Same base:
$$
5x = 100 \Rightarrow x = 20
$$
Check: $5x = 100 > 0$ ✔
Answer: $x = 20$

---

11) $\log_4(2x + 2) = \log_4(3x - 18)$

Same base:
$$
2x + 2 = 3x - 18 \Rightarrow 2 + 18 = 3x - 2x \Rightarrow 20 = x
$$
Check: $2(20)+2 = 42 > 0$, $3(20)-18 = 42 > 0$ ✔
Answer: $x = 20$

---

12) $\log_2(x + 17) = 3\log_2 2$

First simplify right-hand side:
$$
3\log_2 2 = \log_2(2^3) = \log_2(8)
$$
So:
$$
\log_2(x + 17) = \log_2(8) \Rightarrow x + 17 = 8 \Rightarrow x = -9
$$
Check: $x + 17 = -9 + 17 = 8 > 0$ ✔
Answer: $x = -9$

---

✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1) | $x = 4$ |
| 2) | $x = -18$ |
| 3) | $x = 10$ |
| 4) | $x = 6$ |
| 5) | $x = 14$ |
| 6) | $x = 22$ |
| 7) | $x = 19$ |
| 8) | $x = 13$ |
| 9) | $x = 54$ |
| 10) | $x = 20$ |
| 11) | $x = 20$ |
| 12) | $x = -9$ |

All solutions are valid (arguments of logs are positive). ✔

Let me know if you'd like this in a printable format or need explanations for any specific step!