Worksheet with ten exponential and logarithmic equations to solve for x, including examples like 2^(8x-16) - 49 = 67 and 4^(3x+21) - 83 = 2.

Let’s solve each equation step by step. We’ll isolate the exponential part, take logarithms (or natural logs for base e), and solve for x. Then round to 2 decimal places.

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

Add 49 to both sides:
→ $ 2^{8x - 16} = 116 $

Take log of both sides (base 10 or ln — we’ll use ln):
→ $ \ln(2^{8x - 16}) = \ln(116) $

Use power rule: $ \ln(a^b) = b\ln(a) $
→ $ (8x - 16)\ln(2) = \ln(116) $

Divide both sides by $ \ln(2) $:
→ $ 8x - 16 = \frac{\ln(116)}{\ln(2)} $

Calculate right side:
→ $ \ln(116) ≈ 4.7536 $, $ \ln(2) ≈ 0.6931 $
→ $ \frac{4.7536}{0.6931} ≈ 6.858 $

So:
→ $ 8x - 16 = 6.858 $

Add 16:
→ $ 8x = 22.858 $

Divide by 8:
→ $ x ≈ 2.857 → \boxed{2.86} $

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

Add 83:
→ $ 4^{3x + 21} = 85 $

Take ln:
→ $ \ln(4^{3x+21}) = \ln(85) $

→ $ (3x + 21)\ln(4) = \ln(85) $

$ \ln(4) ≈ 1.3863 $, $ \ln(85) ≈ 4.4427 $

→ $ 3x + 21 = \frac{4.4427}{1.3863} ≈ 3.205 $

Subtract 21:
→ $ 3x = 3.205 - 21 = -17.795 $

Divide by 3:
→ $ x ≈ -5.9317 → \boxed{-5.93} $

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

Subtract 33:
→ $ 7^{5x - 65} = 87 $

Take ln:
→ $ (5x - 65)\ln(7) = \ln(87) $

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

→ $ 5x - 65 = \frac{4.4659}{1.9459} ≈ 2.295 $

Add 65:
→ $ 5x = 67.295 $

Divide by 5:
→ $ x ≈ 13.459 → \boxed{13.46} $

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

Add 47:
→ $ 9^{2x - 38} = 170 $

Take ln:
→ $ (2x - 38)\ln(9) = \ln(170) $

$ \ln(9) ≈ 2.1972 $, $ \ln(170) ≈ 5.1358 $

→ $ 2x - 38 = \frac{5.1358}{2.1972} ≈ 2.337 $

Add 38:
→ $ 2x = 40.337 $

Divide by 2:
→ $ x ≈ 20.1685 → \boxed{20.17} $

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

Subtract 46:
→ $ 8^{3x + 39} = 86 $

Take ln:
→ $ (3x + 39)\ln(8) = \ln(86) $

$ \ln(8) ≈ 2.0794 $, $ \ln(86) ≈ 4.4543 $

→ $ 3x + 39 = \frac{4.4543}{2.0794} ≈ 2.142 $

Subtract 39:
→ $ 3x = 2.142 - 39 = -36.858 $

Divide by 3:
→ $ x ≈ -12.286 → \boxed{-12.29} $

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

Add 37:
→ $ 2^{-x - 14} = 24 $

Take ln:
→ $ (-x - 14)\ln(2) = \ln(24) $

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

→ $ -x - 14 = \frac{3.1781}{0.6931} ≈ 4.585 $

Add 14:
→ $ -x = 4.585 + 14 = 18.585 $

Multiply by -1:
→ $ x ≈ -18.585 → \boxed{-18.59} $

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

Add 12:
→ $ e^{4x + 9} = 31 $

Take natural log (ln) — since base is e:
→ $ 4x + 9 = \ln(31) $

$ \ln(31) ≈ 3.4340 $

→ $ 4x = 3.4340 - 9 = -5.566 $

Divide by 4:
→ $ x ≈ -1.3915 → \boxed{-1.39} $

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

Subtract 56:
→ $ e^{x - 15} = 32 $

Take ln:
→ $ x - 15 = \ln(32) $

$ \ln(32) ≈ 3.4657 $

→ $ x = 3.4657 + 15 = 18.4657 → \boxed{18.47} $

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

Add 21:
→ $ 2e^{3x - 11} = 58 $

Divide by 2:
→ $ e^{3x - 11} = 29 $

Take ln:
→ $ 3x - 11 = \ln(29) $

$ \ln(29) ≈ 3.3673 $

→ $ 3x = 3.3673 + 11 = 14.3673 $

Divide by 3:
→ $ x ≈ 4.7891 → \boxed{4.79} $

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

Add 34:
→ $ 3e^{6x - 54} = 117 $

Divide by 3:
→ $ e^{6x - 54} = 39 $

Take ln:
→ $ 6x - 54 = \ln(39) $

$ \ln(39) ≈ 3.6636 $

→ $ 6x = 3.6636 + 54 = 57.6636 $

Divide by 6:
→ $ x ≈ 9.6106 → \boxed{9.61} $

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Final Answer:
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