SOLUTION: Solving Logarithmic Equations Practice Worksheet - Studypool - Free Printable

Let's solve each logarithmic equation step by step and then check the solutions.

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1. $\log_7 x = 3$

#### Solve:
We rewrite the logarithmic equation in exponential form:

$$
\log_b a = c \quad \Leftrightarrow \quad b^c = a
$$

So:
$$
x = 7^3 = 343
$$

✔ Solution: $x = 343$

#### Check:
$$
\log_7(343) = \log_7(7^3) = 3
$$
✔️ Correct.

---

2. $\log_{12} x + 5 = 5$

#### Solve:
Subtract 5 from both sides:
$$
\log_{12} x = 0
$$

Now convert to exponential form:
$$
x = 12^0 = 1
$$

✔ Solution: $x = 1$

#### Check:
$$
\log_{12}(1) + 5 = 0 + 5 = 5
$$
✔️ Correct.

---

3. $4\log_3(x + 8) = 12$

#### Solve:
Divide both sides by 4:
$$
\log_3(x + 8) = 3
$$

Convert to exponential form:
$$
x + 8 = 3^3 = 27
$$

Solve for $x$:
$$
x = 27 - 8 = 19
$$

✔ Solution: $x = 19$

#### Check:
$$
4\log_3(19 + 8) = 4\log_3(27) = 4 \cdot \log_3(3^3) = 4 \cdot 3 = 12
$$
✔️ Correct.

---

4. $\log_2(3x - 2) - 17 = -11$

#### Solve:
Add 17 to both sides:
$$
\log_2(3x - 2) = 6
$$

Convert to exponential form:
$$
3x - 2 = 2^6 = 64
$$

Solve for $x$:
$$
3x = 64 + 2 = 66 \Rightarrow x = \frac{66}{3} = 22
$$

✔ Solution: $x = 22$

#### Check:
$$
\log_2(3(22) - 2) - 17 = \log_2(66 - 2) - 17 = \log_2(64) - 17 = 6 - 17 = -11
$$
✔️ Correct.

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✔ Final Answers:

| Problem | Solution | Check |
|--------|---------|-------|
| 1. $\log_7 x = 3$ | $x = 343$ | $\log_7(343) = 3$ ✔️ |
| 2. $\log_{12} x + 5 = 5$ | $x = 1$ | $\log_{12}(1) + 5 = 5$ ✔️ |
| 3. $4\log_3(x+8) = 12$ | $x = 19$ | $4\log_3(27) = 12$ ✔️ |
| 4. $\log_2(3x-2) - 17 = -11$ | $x = 22$ | $\log_2(64) - 17 = -11$ ✔️ |

All solutions are correct!