Question

part 5: solving exponential equations (unit 3)

1. solve: $5^x = 5^{3x - 10}$
2. solve: $2^{2x + 2} = 8$
3. solve: $3^{3x - 2} = 1$
4. solve: $16^{3x} = 256^{x - 2}$

Explanation:

Problem 19: Solve \( 5^x = 5^{3x - 10} \)

Step 1: Use the property of exponential functions

If \( a^m = a^n \) (where \( a > 0, a
eq 1 \)), then \( m = n \). Here, \( a = 5 \), \( m = x \), and \( n = 3x - 10 \). So we set the exponents equal:
\( x = 3x - 10 \)

Step 2: Solve for \( x \)

Subtract \( x \) from both sides:
\( 0 = 2x - 10 \)
Add 10 to both sides:
\( 10 = 2x \)
Divide both sides by 2:
\( x = 5 \)

Step 1: Rewrite 8 as a power of 2

We know that \( 8 = 2^3 \), so the equation becomes:
\( 2^{2x + 2} = 2^3 \)

Step 2: Use the property of exponential functions

Since the bases are equal (\( a = 2 \)), we set the exponents equal:
\( 2x + 2 = 3 \)

Step 3: Solve for \( x \)

Subtract 2 from both sides:
\( 2x = 1 \)
Divide both sides by 2:
\( x = \frac{1}{2} \)

Step 1: Rewrite 1 as a power of 3

We know that \( 3^0 = 1 \), so the equation becomes:
\( 3^{3x - 2} = 3^0 \)

Step 2: Use the property of exponential functions

Set the exponents equal:
\( 3x - 2 = 0 \)

Step 3: Solve for \( x \)

Add 2 to both sides:
\( 3x = 2 \)
Divide both sides by 3:
\( x = \frac{2}{3} \)

Answer:

\( x = 5 \)

Problem 20: Solve \( 2^{2x + 2} = 8 \)