Worksheet for solving exponential and logarithmic equations.

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

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

Take the logarithm (base 2) of both sides:
$8x - 16 = \log_2(116)$

Calculate $\log_2(116)$:
$\log_2(116) = \frac{\ln(116)}{\ln(2)} \approx \frac{4.7536}{0.6931} \approx 6.867$

So:
$8x - 16 = 6.867$

Add 16 to both sides:
$8x = 22.867$

Divide by 8:
$x = \frac{22.867}{8} \approx 2.86$

$x \approx 2.86$

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

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

Take the logarithm (base 4) of both sides:
$3x + 21 = \log_4(85)$

Calculate $\log_4(85)$:
$\log_4(85) = \frac{\ln(85)}{\ln(4)} \approx \frac{4.4427}{1.3863} \approx 3.205$

So:
$3x + 21 = 3.205$

Subtract 21 from both sides:
$3x = -17.795$

Divide by 3:
$x = \frac{-17.795}{3} \approx -5.93$

$x \approx -5.93$

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

Subtract 33 from both sides:
$7^{5x-65} = 87$

Take the logarithm (base 7) of both sides:
$5x - 65 = \log_7(87)$

Calculate $\log_7(87)$:
$\log_7(87) = \frac{\ln(87)}{\ln(7)} \approx \frac{4.4659}{1.9459} \approx 2.300$

So:
$5x - 65 = 2.300$

Add 65 to both sides:
$5x = 67.300$

Divide by 5:
$x = \frac{67.300}{5} = 13.46$

$x \approx 13.46$

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

Add 47 to both sides:
$9^{2x-38} = 170$

Take the logarithm (base 9) of both sides:
$2x - 38 = \log_9(170)$

Calculate $\log_9(170)$:
$\log_9(170) = \frac{\ln(170)}{\ln(9)} \approx \frac{5.1358}{2.1972} \approx 2.338$

So:
$2x - 38 = 2.338$

Add 38 to both sides:
$2x = 40.338$

Divide by 2:
$x = \frac{40.338}{2} = 20.169$

$x \approx 20.17$

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

Subtract 46 from both sides:
$8^{3x+39} = 86$

Take the logarithm (base 8) of both sides:
$3x + 39 = \log_8(86)$

Calculate $\log_8(86)$:
$\log_8(86) = \frac{\ln(86)}{\ln(8)} \approx \frac{4.4543}{2.0794} \approx 2.144$

So:
$3x + 39 = 2.144$

Subtract 39 from both sides:
$3x = -36.856$

Divide by 3:
$x = \frac{-36.856}{3} \approx -12.29$

$x \approx -12.29$

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

Add 37 to both sides:
$2^{-x-14} = 24$

Take the logarithm (base 2) of both sides:
$-x - 14 = \log_2(24)$

Calculate $\log_2(24)$:
$\log_2(24) = \frac{\ln(24)}{\ln(2)} \approx \frac{3.1781}{0.6931} \approx 4.585$

So:
$-x - 14 = 4.585$

Add 14 to both sides:
$-x = 18.585$

Multiply by -1:
$x = -18.585$

$x \approx -18.59$

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

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

Take the natural logarithm of both sides:
$4x + 9 = \ln(31)$

Calculate $\ln(31)$:
$\ln(31) \approx 3.434$

So:
$4x + 9 = 3.434$

Subtract 9 from both sides:
$4x = -5.566$

Divide by 4:
$x = \frac{-5.566}{4} \approx -1.3915$

$x \approx -1.39$

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

Subtract 56 from both sides:
$e^{x-15} = 32$

Take the natural logarithm of both sides:
$x - 15 = \ln(32)$

Calculate $\ln(32)$:
$\ln(32) = \ln(2^5) = 5\ln(2) \approx 5 \times 0.6931 = 3.4655$

So:
$x - 15 = 3.4655$

Add 15 to both sides:
$x = 18.4655$

$x \approx 18.47$

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

Add 21 to both sides:
$2e^{3x-11} = 58$

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

Take the natural logarithm of both sides:
$3x - 11 = \ln(29)$

Calculate $\ln(29)$:
$\ln(29) \approx 3.367$

So:
$3x - 11 = 3.367$

Add 11 to both sides:
$3x = 14.367$

Divide by 3:
$x = \frac{14.367}{3} \approx 4.789$

$x \approx 4.79$

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

Add 34 to both sides:
$3e^{6x-54} = 117$

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

Take the natural logarithm of both sides:
$6x - 54 = \ln(39)$

Calculate $\ln(39)$:
$\ln(39) \approx 3.6636$

So:
$6x - 54 = 3.6636$

Add 54 to both sides:
$6x = 57.6636$

Divide by 6:
$x = \frac{57.6636}{6} \approx 9.6106$

$x \approx 9.61$