What's Your Strategy? 1. Solve each exponential equation. Explain why you chose the method that you used. a. $$ 2^{x-5}+6=30 $$ b. $$ 7 \cdot(2)^{1 x}=840 $$ C. $$ \mathbf{4}^{x-3}-5=16 $$ d. $$ 10 \cdot\left(\frac{3^{2 x}}{2}\right)=360 $$ e. $$ 2 \cdot 3^{3 x}+1=55 $$

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**a. Solve $$ 2^{x-5}+6=30 $$**  
1. Subtract 6 from both sides:  
$$
2^{x-5} = 30 - 6 = 24
$$  
2. Since the variable is in the exponent, take the logarithm (using any base; here we use natural logarithm):  
$$
x-5 = \frac{\ln(24)}{\ln(2)}
$$  
3. Solve for $$ x $$:  
$$
x = 5 + \frac{\ln(24)}{\ln(2)}
$$  
*Method explanation: We isolated the exponential term and used logarithms to bring the exponent down for solving.*

---

**b. Solve $$ 7\cdot2^{x}=840 $$**  
1. Divide both sides by 7:  
$$
2^{x} = \frac{840}{7} = 120
$$  
2. Take the natural logarithm on both sides:  
$$
x = \frac{\ln(120)}{\ln(2)}
$$  
*Method explanation: After isolating the exponential $$ 2^x $$, logarithms were used because the unknown $$ x $$ appears in the exponent.*

---

**c. Solve $$ 4^{x-3}-5=16 $$**  
1. Add 5 to both sides:  
$$
4^{x-3} = 16+5 = 21
$$  
2. Take the natural logarithm:  
$$
x-3 = \frac{\ln(21)}{\ln(4)}
$$  
3. Solve for $$ x $$:  
$$
x = 3 + \frac{\ln(21)}{\ln(4)}
$$  
*Method explanation: The exponential was isolated, then logarithms were used to extract the variable from the exponent.*

---

**d. Solve $$ 10\cdot\left(\frac{3^{2x}}{2}\right)=360 $$**  
1. Multiply both sides by 2 to eliminate the denominator:  
$$
10\cdot 3^{2x} = 360 \times 2 = 720
$$  
2. Divide both sides by 10:  
$$
3^{2x} = \frac{720}{10} = 72
$$  
3. Take the natural logarithm:  
$$
2x = \frac{\ln(72)}{\ln(3)}
$$  
4. Solve for $$ x $$:  
$$
x = \frac{1}{2}\cdot\frac{\ln(72)}{\ln(3)}
$$  
*Method explanation: Isolating $$ 3^{2x} $$ allowed the use of logarithms to handle the exponent containing the variable.*

---

**e. Solve $$ 2\cdot3^{3x}+1=55 $$**  
1. Subtract 1 from both sides:  
$$
2\cdot3^{3x} = 55-1 = 54
$$  
2. Divide both sides by 2:  
$$
3^{3x} = \frac{54}{2} = 27
$$  
3. Recognize that $$ 27 $$ is a power of 3, namely $$ 27 = 3^3 $$. Therefore,  
$$
3^{3x} = 3^3
$$  
4. Equate the exponents:  
$$
3x = 3
$$  
5. Solve for $$ x $$:  
$$
x = 1
$$  
*Method explanation: By simplifying to reveal the base and comparing exponents, the solution was found directly without needing logarithms.*

**a. $$ x = 5 + \frac{\ln(24)}{\ln(2)} $$** **b. $$ x = \frac{\ln(120)}{\ln(2)} $$** **c. $$ x = 3 + \frac{\ln(21)}{\ln(4)} $$** **d. $$ x = \frac{1}{2}\cdot\frac{\ln(72)}{\ln(3)} $$** **e. $$ x = 1 $$**

What's Your Strategy? 1. Solve each exponential equation. Explain why you chose the method that you used. a. \( 2^{x-5}+6=30 \) b. \( 7 \cdot(2)^{1 x}=840 \) C. \( \mathbf{4}^{x-3}-5=16 \) d. \( 10 \cdot\left(\frac{3^{2 x}}{2}\right)=360 \) e. \( 2 \cdot 3^{3 x}+1=55 \)

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