The required value of x will be -2.7569 .
Given expression,
[tex]4^{x+3} = 7[/tex].
Taking log on both sides we get,
[tex]\rm log\ (4)^{x+3} = log\ 7[/tex]
now applying the logarithmic property, we get
[tex](x+3)\rm log\ 4=log\ 7[/tex] ( [tex]\rm log\ x^m=m\ log\ x[/tex] )
[tex]x+3=\dfrac{\rm log\ 7}{\rm log\ 4}[/tex]
Since,
[tex]\rm\dfrac{ log\ m}{log\ n} =log\ m-log\ n[/tex]
So, [tex]x+3=\rm log\ 7-log\ 4[/tex]
[tex]x+3=0.84509-0.6020[/tex]
[tex]x+3=0.24309[/tex]
[tex]x=-2.75691[/tex]
The nearest thousandth value of x will be -2.7569.
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