Let's begin by looking at the given equation:
[tex]\[ 2^{4x} = 8^{(x-3)} \][/tex]
Our goal is to rewrite this equation so that it has the same base on both sides, which will make it easier to solve and compare.
1. Notice that [tex]\( 8 \)[/tex] can be expressed as a power of [tex]\( 2 \)[/tex]:
[tex]\[ 8 = 2^3 \][/tex]
2. Substitute [tex]\( 8 \)[/tex] with [tex]\( 2^3 \)[/tex] in the equation:
[tex]\[ 8^{(x-3)} = (2^3)^{(x-3)} \][/tex]
3. Use the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex] to simplify the right-hand side:
[tex]\[ (2^3)^{(x-3)} = 2^{3 \cdot (x-3)} = 2^{3x - 9} \][/tex]
4. Now the equation looks like this:
[tex]\[ 2^{4x} = 2^{3x - 9} \][/tex]
The bases on both sides of the equation are now the same. This allows us to equate the exponents directly:
[tex]\[ 4x = 3x - 9 \][/tex]
Therefore, the equivalent equation to [tex]\( 2^{4x} = 8^{(x-3)} \)[/tex] is:
[tex]\[ 2^{4x} = 2^{3x - 9} \][/tex]
So, the correct answer is:
[tex]\[ 2^{4 x}=2^{3 x-9} \][/tex]
