Question

question 16

solve for x in the following equation:

(2^{4x} = 8^{7-x})

• (x = 3)
• (x = -2)
• (x = -3)
• (x = 2)

Explanation:

Express both sides with a common base

Using the base \(2\), rewrite \(8\) as \(2^3\):

$$ 2^{4x} = (2^3)^{7-x} $$

Apply exponent rules to simplify the right side

Multiply the exponents on the right side:

$$ 2^{4x} = 2^{3(7-x)} $$

$$ 2^{4x} = 2^{21-3x} $$

Equate exponents and solve for x

Since the bases are equal, set the exponents equal to each other:

$$ 4x = 21 - 3x $$

$$ 7x = 21 $$

$$ x = 3 $$

Answer:

• (A) x = 3 (Correct answer)
• (B) x = -2
• (C) x = -3
• (D) x = 2