Question

semester exam review
solve the following exponential functions using logarithms. write the exact answer (do not write as decimal!)

1. $2 \cdot 5^{2x} = 50$
2. $8 \cdot 2^{x - 5} = 64$

Explanation:

Problem 1: \( 2 \cdot 5^{2x} = 50 \)

Step 1: Isolate the exponential term

Divide both sides of the equation by 2:
\( \frac{2 \cdot 5^{2x}}{2} = \frac{50}{2} \)
Simplify: \( 5^{2x} = 25 \)

Step 2: Express 25 as a power of 5

Since \( 25 = 5^2 \), substitute into the equation:
\( 5^{2x} = 5^2 \)

Step 3: Set exponents equal (since bases are equal)

For \( a^m = a^n \) (where \( a > 0, a
eq 1 \)), \( m = n \). Thus:
\( 2x = 2 \)

Step 4: Solve for \( x \)

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

Problem 2: \( 8 \cdot 2^{x - 5} = 64 \)

Step 1: Isolate the exponential term

Divide both sides by 8:
\( \frac{8 \cdot 2^{x - 5}}{8} = \frac{64}{8} \)
Simplify: \( 2^{x - 5} = 8 \)

Step 2: Express 8 as a power of 2

Since \( 8 = 2^3 \), substitute into the equation:
\( 2^{x - 5} = 2^3 \)

Step 3: Set exponents equal (since bases are equal)

For \( a^m = a^n \) (where \( a > 0, a
eq 1 \)), \( m = n \). Thus:
\( x - 5 = 3 \)

Step 4: Solve for \( x \)

Add 5 to both sides:
\( x = 3 + 5 = 8 \)

Answer:

s:

1. \( \boldsymbol{x = 1} \)
2. \( \boldsymbol{x = 8} \)