Math worksheet featuring exponential equations to solve using logarithms, with a playful title about lumberjacks and natural logs.

Here are the step-by-step solutions for each equation on the worksheet. We will use logarithms to solve for $x$ in each case.

L. $3^x = 17$
1. Take the natural log ($\ln$) of both sides: $\ln(3^x) = \ln(17)$
2. Bring the exponent down: $x \cdot \ln(3) = \ln(17)$
3. Divide by $\ln(3)$: $x = \frac{\ln(17)}{\ln(3)}$
4. Calculate: $x \approx \frac{2.833}{1.099} \approx 2.58$

M. $20^x = 56$
1. Take the natural log of both sides: $\ln(20^x) = \ln(56)$
2. Bring the exponent down: $x \cdot \ln(20) = \ln(56)$
3. Divide by $\ln(20)$: $x = \frac{\ln(56)}{\ln(20)}$
4. Calculate: $x \approx \frac{4.025}{2.996} \approx 1.34$

R. $e^x = 10$
1. Take the natural log of both sides: $\ln(e^x) = \ln(10)$
2. Since $\ln(e) = 1$, this simplifies to: $x = \ln(10)$
3. Calculate: $x \approx 2.30$

C. $2^{x+3} = 30$
1. Take the natural log of both sides: $\ln(2^{x+3}) = \ln(30)$
2. Bring the exponent down: $(x+3)\ln(2) = \ln(30)$
3. Divide by $\ln(2)$: $x + 3 = \frac{\ln(30)}{\ln(2)}$
4. Calculate the right side: $x + 3 \approx \frac{3.401}{0.693} \approx 4.91$
5. Subtract 3 from both sides: $x \approx 4.91 - 3 = 1.91$

E. $5(6^{3x}) = 20$
1. Divide both sides by 5 first: $6^{3x} = 4$
2. Take the natural log of both sides: $\ln(6^{3x}) = \ln(4)$
3. Bring the exponent down: $3x \cdot \ln(6) = \ln(4)$
4. Isolate $x$: $x = \frac{\ln(4)}{3\ln(6)}$
5. Calculate: $x \approx \frac{1.386}{3(1.792)} \approx \frac{1.386}{5.376} \approx 0.26$

U. $-5 + e^{x+3} = 5$
1. Add 5 to both sides: $e^{x+3} = 10$
2. Take the natural log of both sides: $\ln(e^{x+3}) = \ln(10)$
3. Simplify: $x + 3 = \ln(10)$
4. Calculate $\ln(10)$: $x + 3 \approx 2.30$
5. Subtract 3 from both sides: $x \approx 2.30 - 3 = -0.70$

T. $16^{-x} + 5 = 24$
1. Subtract 5 from both sides: $16^{-x} = 19$
2. Take the natural log of both sides: $\ln(16^{-x}) = \ln(19)$
3. Bring the exponent down: $-x \cdot \ln(16) = \ln(19)$
4. Divide by $-\ln(16)$: $x = \frac{\ln(19)}{-\ln(16)}$
5. Calculate: $x \approx \frac{2.944}{-2.773} \approx -1.06$

N. $3e^{4x+8} = 18$
1. Divide both sides by 3: $e^{4x+8} = 6$
2. Take the natural log of both sides: $\ln(e^{4x+8}) = \ln(6)$
3. Simplify: $4x + 8 = \ln(6)$
4. Calculate $\ln(6)$: $4x + 8 \approx 1.79$
5. Subtract 8 from both sides: $4x \approx 1.79 - 8 = -6.21$
6. Divide by 4: $x \approx \frac{-6.21}{4} \approx -1.55$

Final Answer:
L. x ≈ 2.58
M. x ≈ 1.34
R. x ≈ 2.30
C. x ≈ 1.91
E. x ≈ 0.26
U. x ≈ -0.70
T. x ≈ -1.06
N. x ≈ -1.55