Solving Exponential and Log Equations - Free Printable

Let’s solve each equation step by step. We’ll isolate the exponential part, take logarithms (or natural logs if 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) $
→ $ (8x - 16)\ln(2) = \ln(116) $

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

Calculate:
ln(116) ≈ 4.7536
ln(2) ≈ 0.6931
→ 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:
$ (3x + 21)\ln(4) = \ln(85) $

ln(85) ≈ 4.4427
ln(4) = ln(2²) = 2·ln(2) ≈ 1.3863

So:
$ 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(87) ≈ 4.4659
ln(7) ≈ 1.9459

So:
$ 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(170) ≈ 5.1358
ln(9) = ln(3²) = 2·ln(3) ≈ 2·1.0986 = 2.1972

So:
$ 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(86) ≈ 4.4543
ln(8) = ln(2³) = 3·ln(2) ≈ 3·0.6931 = 2.0793

So:
$ 3x + 39 = \frac{4.4543}{2.0793} ≈ 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(24) ≈ 3.1781
ln(2) ≈ 0.6931

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

Add 14 to both sides? Wait — let’s rearrange:

$ -x = 4.585 + 14 = 18.585 $

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

Wait — check sign:

From:
$ (-x - 14) = 4.585 $
→ $ -x = 4.585 + 14 = 18.585 $
→ $ x = -18.585 → yes, \boxed{-18.59} $

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

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

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

ln(31) ≈ 3.4340

So:
$ 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) = ln(2⁵) = 5·ln(2) ≈ 5·0.6931 = 3.4655

So:
$ x = 3.4655 + 15 = 18.4655 → \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

So:
$ 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

So:
$ 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