Solving Exponential and Log Equations - Free Printable

Let's solve each of the exponential and logarithmic equations step by step. We'll isolate the exponential expression, then use logarithms (natural log or common log) to solve for $ x $. All answers will be rounded to 2 decimal places.

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1. $ 2^{8x - 16} - 49 = 67 $

Step 1: Add 49 to both sides:
$$
2^{8x - 16} = 67 + 49 = 116
$$

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

Step 3: Use log rule: $ \ln(a^b) = b\ln(a) $
$$
(8x - 16)\ln(2) = \ln(116)
$$

Step 4: Solve for $ x $:
$$
8x - 16 = \frac{\ln(116)}{\ln(2)} \approx \frac{5.447}{0.6931} \approx 7.86
$$
$$
8x = 7.86 + 16 = 23.86
$$
$$
x = \frac{23.86}{8} \approx 2.98
$$

✔ Answer: $ x \approx 2.98 $

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2. $ 4^{3x + 21} - 83 = 2 $

Step 1: Add 83 to both sides:
$$
4^{3x + 21} = 85
$$

Step 2: Write 4 as $ 2^2 $, so $ 4^{3x+21} = (2^2)^{3x+21} = 2^{2(3x+21)} = 2^{6x + 42} $

Alternatively, take ln directly:
$$
\ln(4^{3x + 21}) = \ln(85)
$$
$$
(3x + 21)\ln(4) = \ln(85)
$$

Note: $ \ln(4) = \ln(2^2) = 2\ln(2) \approx 2(0.6931) = 1.3862 $

$$
3x + 21 = \frac{\ln(85)}{\ln(4)} \approx \frac{4.4427}{1.3862} \approx 3.207
$$
$$
3x = 3.207 - 21 = -17.793
$$
$$
x = \frac{-17.793}{3} \approx -5.93
$$

✔ Answer: $ x \approx -5.93 $

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3. $ 7^{5x - 65} + 33 = 120 $

Step 1: Subtract 33:
$$
7^{5x - 65} = 87
$$

Step 2: Take ln:
$$
\ln(7^{5x - 65}) = \ln(87)
$$
$$
(5x - 65)\ln(7) = \ln(87)
$$

$ \ln(7) \approx 1.9459 $, $ \ln(87) \approx 4.4659 $

$$
5x - 65 = \frac{4.4659}{1.9459} \approx 2.30
$$
$$
5x = 2.30 + 65 = 67.30
$$
$$
x = \frac{67.30}{5} = 13.46
$$

✔ Answer: $ x \approx 13.46 $

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4. $ 9^{2x - 38} - 47 = 123 $

Step 1: Add 47:
$$
9^{2x - 38} = 170
$$

Step 2: Take ln:
$$
\ln(9^{2x - 38}) = \ln(170)
$$
$$
(2x - 38)\ln(9) = \ln(170)
$$

$ \ln(9) = \ln(3^2) = 2\ln(3) \approx 2(1.0986) = 2.1972 $

$ \ln(170) \approx 5.1356 $

$$
2x - 38 = \frac{5.1356}{2.1972} \approx 2.337
$$
$$
2x = 2.337 + 38 = 40.337
$$
$$
x = \frac{40.337}{2} \approx 20.17
$$

✔ Answer: $ x \approx 20.17 $

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5. $ 8^{3x + 39} + 46 = 132 $

Step 1: Subtract 46:
$$
8^{3x + 39} = 86
$$

Step 2: Take ln:
$$
\ln(8^{3x + 39}) = \ln(86)
$$
$$
(3x + 39)\ln(8) = \ln(86)
$$

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

$ \ln(86) \approx 4.4543 $

$$
3x + 39 = \frac{4.4543}{2.0794} \approx 2.142
$$
$$
3x = 2.142 - 39 = -36.858
$$
$$
x = \frac{-36.858}{3} \approx -12.29
$$

✔ Answer: $ x \approx -12.29 $

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6. $ 2^{-x - 14} - 37 = -13 $

Step 1: Add 37:
$$
2^{-x - 14} = 24
$$

Step 2: Take ln:
$$
\ln(2^{-x - 14}) = \ln(24)
$$
$$
(-x - 14)\ln(2) = \ln(24)
$$

$ \ln(2) \approx 0.6931 $, $ \ln(24) \approx 3.1781 $

$$
-x - 14 = \frac{3.1781}{0.6931} \approx 4.585
$$
$$
-x = 4.585 + 14 = 18.585
$$
$$
x = -18.585 \approx -18.59
$$

✔ Answer: $ x \approx -18.59 $

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7. $ e^{4x + 9} - 12 = 19 $

Step 1: Add 12:
$$
e^{4x + 9} = 31
$$

Step 2: Take natural log (ln):
$$
\ln(e^{4x + 9}) = \ln(31)
$$
$$
4x + 9 = \ln(31) \approx 3.4339
$$
$$
4x = 3.4339 - 9 = -5.5661
$$
$$
x = \frac{-5.5661}{4} \approx -1.39
$$

✔ Answer: $ x \approx -1.39 $

---

8. $ e^{x - 15} + 56 = 88 $

Step 1: Subtract 56:
$$
e^{x - 15} = 32
$$

Step 2: Take ln:
$$
\ln(e^{x - 15}) = \ln(32)
$$
$$
x - 15 = \ln(32) \approx 3.4657
$$
$$
x = 3.4657 + 15 = 18.4657 \approx 18.47
$$

✔ Answer: $ x \approx 18.47 $

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9. $ 2e^{3x - 11} - 21 = 37 $

Step 1: Add 21:
$$
2e^{3x - 11} = 58
$$

Step 2: Divide by 2:
$$
e^{3x - 11} = 29
$$

Step 3: Take ln:
$$
\ln(e^{3x - 11}) = \ln(29)
$$
$$
3x - 11 = \ln(29) \approx 3.3673
$$
$$
3x = 3.3673 + 11 = 14.3673
$$
$$
x = \frac{14.3673}{3} \approx 4.79
$$

✔ Answer: $ x \approx 4.79 $

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10. $ 3e^{6x - 54} - 34 = 83 $

Step 1: Add 34:
$$
3e^{6x - 54} = 117
$$

Step 2: Divide by 3:
$$
e^{6x - 54} = 39
$$

Step 3: Take ln:
$$
\ln(e^{6x - 54}) = \ln(39)
$$
$$
6x - 54 = \ln(39) \approx 3.6636
$$
$$
6x = 3.6636 + 54 = 57.6636
$$
$$
x = \frac{57.6636}{6} \approx 9.61
$$

✔ Answer: $ x \approx 9.61 $

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✔ Final Answers (Rounded to 2 Decimal Places):

1. $ x \approx 2.98 $
2. $ x \approx -5.93 $
3. $ x \approx 13.46 $
4. $ x \approx 20.17 $
5. $ x \approx -12.29 $
6. $ x \approx -18.59 $
7. $ x \approx -1.39 $
8. $ x \approx 18.47 $
9. $ x \approx 4.79 $
10. $ x \approx 9.61 $

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