Answer:
Question (a)
Given equation:
[tex]8^{2x} = 32^{x+3}[/tex]
8 can be written as [tex]2^3[/tex]
32 can be written as [tex]2^5[/tex]
Therefore, we can rewrite the equation with base 2:
[tex]\implies (2^3)^{2x} = (2^5)^{x+3}[/tex]
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Question (b)
To solve:
[tex](2^3)^{2x} = (2^5)^{x+3}[/tex]
Apply the exponent rule [tex](a^b)^c=a^{bc}[/tex] :
[tex]\implies 2^{3 \cdot 2x} = 2^{5(x+3)}[/tex]
[tex]\implies 2^{6x} = 2^{5x+15}[/tex]
[tex]\textsf{If }a^{f(x)}=a^{g(x)}, \textsf{ then } f(x)=g(x)[/tex] :
[tex]\implies 6x = 5x+15[/tex]
Subtract [tex]5x[/tex] from both sides:
[tex]\implies x = 15[/tex]
