Remember to show work and explain. Use the math font. [Logarithms, Algebra 2] Find the inverse of each function: 1. \: y = 3 \: log_(3) \: {x}^(4) 2. \: y = log_(3) \: {x}^(4) 3…

Question

Remember to show work and explain. Use the math font.

[Logarithms, Algebra 2]

Find the inverse of each function:


`1. \: y = 3 \: log_(3) \: {x}^(4)`

`2. \: y = log_(3) \: {x}^(4)`

`3. \: y = - log_(4) \: {x}^(3) + 7`
​

Answer 1

`\large\boxed{1.\ f^(-1)(x)=\sqrt[12]{3^x}}\\\\\boxed{2.\ f^(-1)(x)=\sqrt[4]{3^x}}\\\\\ \boxed{3.\ f^(-1)(x)=\sqrt[3]{4^(7-x)}}`

Explanation:

`(a^n)^m=a^(nm)\\\\\log_ab=c\iff a^c=b\\\\a^(\log_ax)=x\\\\n\log_ab=\log_ab^n\\\\\log_ab+\log_ac=\log_a(bc)\\============================\\\\1.\\y=3\log_3x^4\to y=\log_3(x^4)^3\to y=\log_3x^(12)`

`2.\\y=\log_3x^4\\\\\text{Exchange x and y. Solve for y:}\\\\\log_3y^4=x\Rightarrow3^(\log_3y^4)=3^x\Rightarrow y^(4)=3^x\\\\y=\sqrt[4]{3^x}\\-------------------------`

`3.\\y=-\log_4x^3+7\\\\\text{Exchange x and y. Solve for y:}\\\\-\log_4y^3+7=x\qquad\text{subtract 7 from both sides}\\\\-\log_4 y^3=x-7\qquad\text{change the signs}\\\\\log_4y^3=7-x\Rightarrow4^(\log_4y^3)=4^(7-x)\\\\y^3=4^(7-x)\Rightarrow y=\sqrt[3]{4^(7-x)}`