Answer:
Z = 0.198877274
Step-by-step explanation:
[tex](\frac{1}{4})^{3z-1} = 16^z + 2*16^{z-2}\\4^{1-3z} = 4^{2z} + 4^{\frac{1}{2}}*4^{2z-4}\\4^{1-3z} = 4^{2z} + 4^{2z-4+\frac{1}{2}}\\4^{1-3z} = 4^{2z} + 4^{2z-\frac{7}{2}}\\4^{1-3z} = 4^{2z} *(1+ 4^{-\frac{7}{2}})\\4^{1-3z} = 4^{2z} *(1+ 2^{-7})\\4^{1-3z} = 4^{2z} *(1+ \frac{1}{128} )\\4^{1-3z} = 4^{2z} *(\frac{129}{128} )\\Taking\;\; Logarithm\;\; with\;\; base\;\; 4\\Log_4(4^{1-3z}) = Log_4(4^{2z}) + Log_4(\frac{129}{128})\\1-3z = 2z + 0.005613627712 \\5z = 0.994386372\\z = 0.198877274[/tex]
Hence, the value of Z = 0.198877274
Answer:
The value of z is [tex]\frac{3}{8}[/tex]
Step-by-step explanation:
Given equation,
[tex](\frac{1}{4})^{3z-1}=16^{z+2}.64^{z-2}[/tex]
[tex]\frac{1}{4^{3z-1}}=(4)^{2z+4}.(4)^{3z-6}[/tex]
[tex]4^{1-3z}=4^{2z+4+3z-6}[/tex] [tex](a^m.a^n=a^{m+n}\text{ and }a^m=\frac{1}{a^{-m}})[/tex]
[tex]4^{1-3z}=4^{5z-2}[/tex]
By comparing the exponents,
[tex]1-3z=5z-2[/tex]
[tex]-3z-5z=-2-1[/tex]
[tex]-8z=-3[/tex]
[tex]\implies z=\frac{3}{8}[/tex]
