Question

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

[Logarithms, Algebra 2]

Find the inverse of each function:


`1. \:y=\: (\frac{ {5}^(x) }{2})^{ (1)/(4) }`

`2. \: y = ( {10}^(x) - 5) ^{ (1)/(5) }`

`3. \: y = \: log_(4)(4 {x}^(2) )`
​

Answer 1

`\large\boxed{1.\ f^(-1)(x)=4\log(x\sqrt[4]2)}\\\\\boxed{2.\ f^(-1)(x)=\log(x^5+5)}\\\\\boxed{3.\ f^(-1)(x)=\sqrt{4^(x-1)}}`

Explanation:

`\log_ab=c\iff a^c=b\\\\n\log_ab=\log_ab^n\\\\a^(\log_ab)=b\\\\\log_aa^n=n\\\\\log_(10)a=\log a\\=============================`

`1.\\y=\left((5^x)/(2)\right)^(1)/(4)\\\\\text{Exchange x and y. Solve for y:}\\\\\left((5^y)/(2)\right)^(1)/(4)=x\qquad\text{use}\ \left((a)/(b)\right)^n=(a^n)/(b^n)\\\\((5^y)^(1)/(4))/(2^(1)/(4))=x\qquad\text{multiply both sides by }\ 2^(1)/(4)\\\\\left(5^y\right)^(1)/(4)=2^(1)/(4)x\qquad\text{use}\ (a^n)^m=a^(nm)\\\\5^{(1)/(4)y}=2^(1)/(4)x\qquad\log_5\ \text{of both sides}`

`\log_55^{(1)/(4)y}=\log_5\left(2^(1)/(4)x\right)\qquad\text{use}\ a^(1)/(n)=\sqrt[n]{a}\\\\(1)/(4)y=\log(x\sqrt[4]2)\qquad\text{multiply both sides by 4}\\\\y=4\log(x\sqrt[4]2)`

`--------------------------\\2.\\y=(10^x-5)^(1)/(5)\\\\\text{Exchange x and y. Solve for y:}\\\\(10^y-5)^(1)/(5)=x\qquad\text{5 power of both sides}\\\\\bigg[(10^y-5)^(1)/(5)\bigg]^5=x^5\qquad\text{use}\ (a^n)^m=a^(nm)\\\\(10^y-5)^{(1)/(5)\cdot5}=x^5\\\\10^y-5=x^5\qquad\text{add 5 to both sides}\\\\10^y=x^5+5\qquad\log\ \text{of both sides}\\\\\log10^y=\log(x^5+5)\Rightarrow y=\log(x^5+5)`

`--------------------------\\3.\\y=\log_4(4x^2)\\\\\text{Exchange x and y. Solve for y:}\\\\\log_4(4y^2)=x\Rightarrow4^(\log_4(4y^2))=4^x\\\\4y^2=4^x\qquad\text{divide both sides by 4}\\\\y^2=(4^x)/(4)\qquad\text{use}\ (a^n)/(a^m)=a^(n-m)\\\\y^2=4^(x-1)\Rightarrow y=\sqrt{4^(x-1)}`