Worksheet for solving exponential equations using logarithms, including examples like 3^x = 17 and 12^x = 13.

Let's solve each of these exponential equations using logarithms. We'll use the logarithmic property:

> If $ a^x = b $, then $ x = \log_a(b) $, or equivalently, $ x = \frac{\log(b)}{\log(a)} $.

We'll apply this method to each equation and round answers to the nearest ten-thousandth (i.e., 4 decimal places).

---

1) $ 3^x = 17 $

Take log of both sides:
$$
x = \log_3(17) = \frac{\log(17)}{\log(3)}
$$

Using calculator:
- $ \log(17) \approx 1.2304 $
- $ \log(3) \approx 0.4771 $

$$
x \approx \frac{1.2304}{0.4771} \approx 2.5811
$$

✔ Answer: $ x \approx 2.5811 $

---

2) $ 12^x = 13 $

$$
x = \log_{12}(13) = \frac{\log(13)}{\log(12)}
$$

- $ \log(13) \approx 1.1139 $
- $ \log(12) \approx 1.0792 $

$$
x \approx \frac{1.1139}{1.0792} \approx 1.0321
$$

✔ Answer: $ x \approx 1.0321 $

---

3) $ 9^x = 49 $

Note: $ 9 = 3^2 $, but we can just use logs directly.

$$
x = \log_9(49) = \frac{\log(49)}{\log(9)}
$$

- $ \log(49) = \log(7^2) = 2\log(7) \approx 2 \times 0.8451 = 1.6902 $
- $ \log(9) = \log(3^2) = 2\log(3) \approx 2 \times 0.4771 = 0.9542 $

$$
x \approx \frac{1.6902}{0.9542} \approx 1.7707
$$

✔ Answer: $ x \approx 1.7707 $

---

4) $ 16^x = 67 $

$$
x = \log_{16}(67) = \frac{\log(67)}{\log(16)}
$$

- $ \log(67) \approx 1.8261 $
- $ \log(16) = \log(2^4) = 4\log(2) \approx 4 \times 0.3010 = 1.2040 $

$$
x \approx \frac{1.8261}{1.2040} \approx 1.5167
$$

✔ Answer: $ x \approx 1.5167 $

---

5) $ 3^x = 69 $

$$
x = \log_3(69) = \frac{\log(69)}{\log(3)}
$$

- $ \log(69) \approx 1.8388 $
- $ \log(3) \approx 0.4771 $

$$
x \approx \frac{1.8388}{0.4771} \approx 3.8556
$$

✔ Answer: $ x \approx 3.8556 $

---

6) $ 6^x = 51 $

$$
x = \log_6(51) = \frac{\log(51)}{\log(6)}
$$

- $ \log(51) \approx 1.7095 $
- $ \log(6) = \log(2 \cdot 3) = \log(2) + \log(3) \approx 0.3010 + 0.4771 = 0.7781 $

$$
x \approx \frac{1.7095}{0.7781} \approx 2.2000
$$

✔ Answer: $ x \approx 2.2000 $

---

7) $ 6^x = 99 $

$$
x = \log_6(99) = \frac{\log(99)}{\log(6)}
$$

- $ \log(99) \approx 1.9972 $
- $ \log(6) \approx 0.7781 $

$$
x \approx \frac{1.9972}{0.7781} \approx 2.5677
$$

✔ Answer: $ x \approx 2.5677 $

---

8) $ 20^x = 56 $

$$
x = \log_{20}(56) = \frac{\log(56)}{\log(20)}
$$

- $ \log(56) \approx 1.7482 $
- $ \log(20) = \log(2 \cdot 10) = \log(2) + \log(10) \approx 0.3010 + 1 = 1.3010 $

$$
x \approx \frac{1.7482}{1.3010} \approx 1.3437
$$

✔ Answer: $ x \approx 1.3437 $

---

9) $ 3 \cdot 18^{6x} = 26 $

First, isolate the exponential term:

$$
18^{6x} = \frac{26}{3} \approx 8.6667
$$

Now take logs:

$$
6x = \log_{18}(8.6667) = \frac{\log(8.6667)}{\log(18)}
$$

- $ \log(8.6667) \approx 0.9365 $
- $ \log(18) = \log(2 \cdot 3^2) = \log(2) + 2\log(3) \approx 0.3010 + 2(0.4771) = 1.2552 $

$$
6x \approx \frac{0.9365}{1.2552} \approx 0.7462
$$

$$
x \approx \frac{0.7462}{6} \approx 0.1244
$$

✔ Answer: $ x \approx 0.1244 $

---

10) $ e^{x - 1} - 5 = 3 $

Solve for $ e^{x - 1} $:

$$
e^{x - 1} = 8
$$

Take natural log (ln):

$$
x - 1 = \ln(8)
$$

$$
x = \ln(8) + 1
$$

- $ \ln(8) = \ln(2^3) = 3\ln(2) \approx 3 \times 0.6931 = 2.0794 $

$$
x \approx 2.0794 + 1 = 3.0794
$$

✔ Answer: $ x \approx 3.0794 $

---

11) $ 9^{x + 10} + 3 = 81 $

Subtract 3:

$$
9^{x + 10} = 78
$$

Now solve:

$$
x + 10 = \log_9(78) = \frac{\log(78)}{\log(9)}
$$

- $ \log(78) \approx 1.8975 $
- $ \log(9) \approx 0.9542 $

$$
x + 10 \approx \frac{1.8975}{0.9542} \approx 1.9883
$$

$$
x \approx 1.9883 - 10 = -8.0117
$$

✔ Answer: $ x \approx -8.0117 $

---

12) $ 11^{x - 8} - 5 = 54 $

Add 5:

$$
11^{x - 8} = 59
$$

Take log:

$$
x - 8 = \log_{11}(59) = \frac{\log(59)}{\log(11)}
$$

- $ \log(59) \approx 1.7709 $
- $ \log(11) \approx 1.0414 $

$$
x - 8 \approx \frac{1.7709}{1.0414} \approx 1.6998
$$

$$
x \approx 1.6998 + 8 = 9.6998
$$

✔ Answer: $ x \approx 9.6998 $

---

✔ Final Answers (Rounded to Nearest Ten-Thousandth):

| Problem | Answer |
|--------|--------|
| 1) | 2.5811 |
| 2) | 1.0321 |
| 3) | 1.7707 |
| 4) | 1.5167 |
| 5) | 3.8556 |
| 6) | 2.2000 |
| 7) | 2.5677 |
| 8) | 1.3437 |
| 9) | 0.1244 |
| 10) | 3.0794 |
| 11) | -8.0117 |
| 12) | 9.6998 |

Let me know if you'd like a step-by-step explanation for any specific problem!