Worksheet with ten exponential and logarithmic equations to solve for x, requiring rounding to two decimal places and showing work.

Problem: Solve each equation for \( x \). Round to 2 decimal places and show work.

We will solve each equation step by step.

---

#### 1. \( 2^{8x-16} - 49 = 67 \)

1. Add 49 to both sides:
\[
2^{8x-16} = 116
\]

2. Take the natural logarithm (ln) of both sides:
\[
\ln(2^{8x-16}) = \ln(116)
\]

3. Use the logarithmic property \( \ln(a^b) = b \ln(a) \):
\[
(8x - 16) \ln(2) = \ln(116)
\]

4. Solve for \( 8x - 16 \):
\[
8x - 16 = \frac{\ln(116)}{\ln(2)}
\]

5. Calculate \( \ln(116) \) and \( \ln(2) \):
\[
\ln(116) \approx 4.754, \quad \ln(2) \approx 0.693
\]
\[
8x - 16 = \frac{4.754}{0.693} \approx 6.86
\]

6. Solve for \( x \):
\[
8x = 6.86 + 16
\]
\[
8x = 22.86
\]
\[
x = \frac{22.86}{8} \approx 2.86
\]

Answer:
\[
\boxed{2.86}
\]

---

#### 2. \( 4^{3x+21} - 83 = 2 \)

1. Add 83 to both sides:
\[
4^{3x+21} = 85
\]

2. Take the natural logarithm (ln) of both sides:
\[
\ln(4^{3x+21}) = \ln(85)
\]

3. Use the logarithmic property \( \ln(a^b) = b \ln(a) \):
\[
(3x + 21) \ln(4) = \ln(85)
\]

4. Solve for \( 3x + 21 \):
\[
3x + 21 = \frac{\ln(85)}{\ln(4)}
\]

5. Calculate \( \ln(85) \) and \( \ln(4) \):
\[
\ln(85) \approx 4.443, \quad \ln(4) \approx 1.386
\]
\[
3x + 21 = \frac{4.443}{1.386} \approx 3.205
\]

6. Solve for \( x \):
\[
3x = 3.205 - 21
\]
\[
3x = -17.795
\]
\[
x = \frac{-17.795}{3} \approx -5.93
\]

Answer:
\[
\boxed{-5.93}
\]

---

#### 3. \( 7^{5x-65} + 33 = 120 \)

1. Subtract 33 from both sides:
\[
7^{5x-65} = 87
\]

2. Take the natural logarithm (ln) of both sides:
\[
\ln(7^{5x-65}) = \ln(87)
\]

3. Use the logarithmic property \( \ln(a^b) = b \ln(a) \):
\[
(5x - 65) \ln(7) = \ln(87)
\]

4. Solve for \( 5x - 65 \):
\[
5x - 65 = \frac{\ln(87)}{\ln(7)}
\]

5. Calculate \( \ln(87) \) and \( \ln(7) \):
\[
\ln(87) \approx 4.466, \quad \ln(7) \approx 1.946
\]
\[
5x - 65 = \frac{4.466}{1.946} \approx 2.295
\]

6. Solve for \( x \):
\[
5x = 2.295 + 65
\]
\[
5x = 67.295
\]
\[
x = \frac{67.295}{5} \approx 13.46
\]

Answer:
\[
\boxed{13.46}
\]

---

#### 4. \( 9^{2x-38} - 47 = 123 \)

1. Add 47 to both sides:
\[
9^{2x-38} = 170
\]

2. Take the natural logarithm (ln) of both sides:
\[
\ln(9^{2x-38}) = \ln(170)
\]

3. Use the logarithmic property \( \ln(a^b) = b \ln(a) \):
\[
(2x - 38) \ln(9) = \ln(170)
\]

4. Solve for \( 2x - 38 \):
\[
2x - 38 = \frac{\ln(170)}{\ln(9)}
\]

5. Calculate \( \ln(170) \) and \( \ln(9) \):
\[
\ln(170) \approx 5.136, \quad \ln(9) \approx 2.197
\]
\[
2x - 38 = \frac{5.136}{2.197} \approx 2.338
\]

6. Solve for \( x \):
\[
2x = 2.338 + 38
\]
\[
2x = 40.338
\]
\[
x = \frac{40.338}{2} \approx 20.17
\]

Answer:
\[
\boxed{20.17}
\]

---

#### 5. \( 8^{3x+39} + 46 = 132 \)

1. Subtract 46 from both sides:
\[
8^{3x+39} = 86
\]

2. Take the natural logarithm (ln) of both sides:
\[
\ln(8^{3x+39}) = \ln(86)
\]

3. Use the logarithmic property \( \ln(a^b) = b \ln(a) \):
\[
(3x + 39) \ln(8) = \ln(86)
\]

4. Solve for \( 3x + 39 \):
\[
3x + 39 = \frac{\ln(86)}{\ln(8)}
\]

5. Calculate \( \ln(86) \) and \( \ln(8) \):
\[
\ln(86) \approx 4.454, \quad \ln(8) \approx 2.079
\]
\[
3x + 39 = \frac{4.454}{2.079} \approx 2.142
\]

6. Solve for \( x \):
\[
3x = 2.142 - 39
\]
\[
3x = -36.858
\]
\[
x = \frac{-36.858}{3} \approx -12.29
\]

Answer:
\[
\boxed{-12.29}
\]

---

#### 6. \( 2^{-x-14} - 37 = -13 \)

1. Add 37 to both sides:
\[
2^{-x-14} = 24
\]

2. Take the natural logarithm (ln) of both sides:
\[
\ln(2^{-x-14}) = \ln(24)
\]

3. Use the logarithmic property \( \ln(a^b) = b \ln(a) \):
\[
(-x - 14) \ln(2) = \ln(24)
\]

4. Solve for \( -x - 14 \):
\[
-x - 14 = \frac{\ln(24)}{\ln(2)}
\]

5. Calculate \( \ln(24) \) and \( \ln(2) \):
\[
\ln(24) \approx 3.178, \quad \ln(2) \approx 0.693
\]
\[
-x - 14 = \frac{3.178}{0.693} \approx 4.586
\]

6. Solve for \( x \):
\[
-x = 4.586 + 14
\]
\[
-x = 18.586
\]
\[
x = -18.59
\]

Answer:
\[
\boxed{-18.59}
\]

---

#### 7. \( e^{4x+9} - 12 = 19 \)

1. Add 12 to both sides:
\[
e^{4x+9} = 31
\]

2. Take the natural logarithm (ln) of both sides:
\[
\ln(e^{4x+9}) = \ln(31)
\]

3. Use the logarithmic property \( \ln(e^b) = b \):
\[
4x + 9 = \ln(31)
\]

4. Solve for \( 4x \):
\[
4x = \ln(31) - 9
\]

5. Calculate \( \ln(31) \):
\[
\ln(31) \approx 3.434
\]
\[
4x = 3.434 - 9
\]
\[
4x = -5.566
\]

6. Solve for \( x \):
\[
x = \frac{-5.566}{4} \approx -1.39
\]

Answer:
\[
\boxed{-1.39}
\]

---

#### 8. \( e^{x-15} + 56 = 88 \)

1. Subtract 56 from both sides:
\[
e^{x-15} = 32
\]

2. Take the natural logarithm (ln) of both sides:
\[
\ln(e^{x-15}) = \ln(32)
\]

3. Use the logarithmic property \( \ln(e^b) = b \):
\[
x - 15 = \ln(32)
\]

4. Solve for \( x \):
\[
x = \ln(32) + 15
\]

5. Calculate \( \ln(32) \):
\[
\ln(32) \approx 3.466
\]
\[
x = 3.466 + 15
\]
\[
x = 18.47
\]

Answer:
\[
\boxed{18.47}
\]

---

#### 9. \( 2e^{3x-11} - 21 = 37 \)

1. Add 21 to both sides:
\[
2e^{3x-11} = 58
\]

2. Divide both sides by 2:
\[
e^{3x-11} = 29
\]

3. Take the natural logarithm (ln) of both sides:
\[
\ln(e^{3x-11}) = \ln(29)
\]

4. Use the logarithmic property \( \ln(e^b) = b \):
\[
3x - 11 = \ln(29)
\]

5. Solve for \( 3x \):
\[
3x = \ln(29) + 11
\]

6. Calculate \( \ln(29) \):
\[
\ln(29) \approx 3.367
\]
\[
3x = 3.367 + 11
\]
\[
3x = 14.367
\]

7. Solve for \( x \):
\[
x = \frac{14.367}{3} \approx 4.79
\]

Answer:
\[
\boxed{4.79}
\]

---

#### 10. \( 3e^{6x-54} - 34 = 83 \)

1. Add 34 to both sides:
\[
3e^{6x-54} = 117
\]

2. Divide both sides by 3:
\[
e^{6x-54} = 39
\]

3. Take the natural logarithm (ln) of both sides:
\[
\ln(e^{6x-54}) = \ln(39)
\]

4. Use the logarithmic property \( \ln(e^b) = b \):
\[
6x - 54 = \ln(39)
\]

5. Solve for \( 6x \):
\[
6x = \ln(39) + 54
\]

6. Calculate \( \ln(39) \):
\[
\ln(39) \approx 3.664
\]
\[
6x = 3.664 + 54
\]
\[
6x = 57.664
\]

7. Solve for \( x \):
\[
x = \frac{57.664}{6} \approx 9.61
\]

Answer:
\[
\boxed{9.61}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ 2.86 \\
2. & \ -5.93 \\
3. & \ 13.46 \\
4. & \ 20.17 \\
5. & \ -12.29 \\
6. & \ -18.59 \\
7. & \ -1.39 \\
8. & \ 18.47 \\
9. & \ 4.79 \\
10. & \ 9.61
\end{aligned}
}
\]