Question

If (2x-4)^12=[(4)²]^6 , find x​

Answer 1

Answer:

$\displaystyle{\boxed{\red{\sf\:x\:=\:4\:}}}$

Step-by-step-explanation:

We have given that,

$\displaystyle{\sf\:(\:2x\:-\:4\:)^{12}\:=\:[\:(\:4\:)^2\:]^6}$

We have to find the value of x.

Now,

$\displaystyle{\sf\:(\:2x\:-\:4\:)^{12}\:=\:[\:(\:4\:)^2\:]^6}$

$\displaystyle{\implies\sf\:(\:2x\:-\:4\:)^{12}\:=\:(\:4\:)^{2\:\times\:6}\:\qquad\cdots[\:(\:a^m\:)^n\:=\:a^{m\:\times\:n}\:]}$

$\displaystyle{\implies\sf\:(\:2x\:-\:4\:)^{12}\:=\:4^{12}}$

Taking log to the base 4 on both sides, we get,

$\displaystyle{\implies\sf\:\log_4\:(\:2x\:-\:4\:)^{12}\:=\:\log_4\:(\:4^{12}\:)}$

We know that,

$\displaystyle{\pink{\sf\:\log_b\:(\:b^k\:)\:=\:k}}$

$\displaystyle{\implies\sf\:\log_4\:(\:2x\:-\:4\:)^{12}\:=\:12}$

We know that,

$\displaystyle{\pink{\sf\:\log_b\:(\:a^k\:)\:=\:k\:\log_b\:(\:a\:)}}$

$\displaystyle{\implies\sf\:12\:\log_4\:(\:2x\:-\:4\:)\:=\:12}$

$\displaystyle{\implies\sf\:\log_4\:(\:2x\:-\:4\:)\:=\:\cancel{\dfrac{12}{12}}}$

$\displaystyle{\implies\sf\:\log_4\:(\:2x\:-\:4\:)\:=\:1}$

We know that,

$\displaystyle{\pink{\sf\:\log_b\:(\:b\:)\:=\:1}}$

$\displaystyle{\implies\sf\:\log_4\:(\:2x\:-\:4\:)\:=\:\log_4\:(\:4\:)}$

Taking antilog to the base 4 on both sides, we get,

$\displaystyle{\implies\sf\:2x\:-\:4\:=\:4}$

$\displaystyle{\implies\sf\:2x\:=\:4\:+\:4}$

$\displaystyle{\implies\sf\:2x\:=\:8}$

$\displaystyle{\implies\sf\:x\:=\:\cancel{\dfrac{8}{2}}}$

$\displaystyle{\implies\underline{\boxed{\red{\sf\:x\:=\:4\:}}}}$

Answer 2

Answer:

$\huge\color{red}{ \colorbox{green}{\colorbox{aqua} {x = 4}}}$

Step-by-step explanation:

It's in the attachment,

have an glorious day ahead !!!