Question

solve the following by rewriting each side as powers of the same base.
$625^x = 125^{x - 1}$
$x = \square$
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solve the following equation for n.
$3(4)^n = 48$
$n = \square$
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Explanation:

First Problem: Solve \( 625^x = 125^{x - 1} \)

Step1: Express bases as powers of 5

\( 625 = 5^4 \) and \( 125 = 5^3 \), so rewrite the equation:
\( (5^4)^x = (5^3)^{x - 1} \)

Step2: Simplify exponents (power of a power)

Using \( (a^m)^n = a^{mn} \), we get:
\( 5^{4x} = 5^{3(x - 1)} \)

Step3: Set exponents equal (since bases are equal, exponents must be equal)

\( 4x = 3(x - 1) \)

Step4: Solve for \( x \)

Expand the right side: \( 4x = 3x - 3 \)
Subtract \( 3x \) from both sides: \( x = -3 \)

Step1: Isolate the exponential term

Divide both sides by 3:
\( 4^n = \frac{48}{3} = 16 \)

Step2: Express 16 as a power of 4

\( 16 = 4^2 \), so rewrite the equation:
\( 4^n = 4^2 \)

Step3: Set exponents equal (since bases are equal, exponents must be equal)

\( n = 2 \)

Answer:

\( x = -3 \)

Second Problem: Solve \( 3(4)^n = 48 \)