Need help please find the inverse of this logarithm function please ❤️ ONLY IF UR GOOD AT MATH!!!!
2 answers:
as you already know, we start by doing a quick switcheroo on the variables, to get the inverse expression of any expression, so
![\bf \textit{Logarithm of exponentials} \\\\ \log_a\left( x^b \right)\implies b\cdot \log_a(x) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \stackrel{f(x)}{y}=3\left( 2^{x+3} \right)-4\implies \stackrel{\textit{quick switcheroo}}{x=3\left( 2^{y+3} \right)-4}\implies x+4=3\left( 2^{y+3} \right)](https://tex.z-dn.net/?f=%5Cbf%20%5Ctextit%7BLogarithm%20of%20exponentials%7D%20%5C%5C%5C%5C%20%5Clog_a%5Cleft%28%20x%5Eb%20%5Cright%29%5Cimplies%20b%5Ccdot%20%5Clog_a%28x%29%20%5C%5C%5C%5C%5B-0.35em%5D%20%5Crule%7B34em%7D%7B0.25pt%7D%5C%5C%5C%5C%20%5Cstackrel%7Bf%28x%29%7D%7By%7D%3D3%5Cleft%28%202%5E%7Bx%2B3%7D%20%5Cright%29-4%5Cimplies%20%5Cstackrel%7B%5Ctextit%7Bquick%20switcheroo%7D%7D%7Bx%3D3%5Cleft%28%202%5E%7By%2B3%7D%20%5Cright%29-4%7D%5Cimplies%20x%2B4%3D3%5Cleft%28%202%5E%7By%2B3%7D%20%5Cright%29)
![\bf \cfrac{x+4}{3}=2^{y+3}\implies \log\left( \cfrac{x+4}{3} \right)=\log\left( 2^{y+3} \right) \\\\\\ \log\left( \cfrac{x+4}{3} \right)=(y+3)\log\left( 2 \right)\implies \cfrac{\log\left( \frac{x+4}{3} \right)}{\log(2)}=y+3 \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill \cfrac{\log\left( \frac{x+4}{3} \right)}{\log(2)}-3=\stackrel{f^{-1}(x)}{y}~\hfill](https://tex.z-dn.net/?f=%5Cbf%20%5Ccfrac%7Bx%2B4%7D%7B3%7D%3D2%5E%7By%2B3%7D%5Cimplies%20%5Clog%5Cleft%28%20%5Ccfrac%7Bx%2B4%7D%7B3%7D%20%5Cright%29%3D%5Clog%5Cleft%28%202%5E%7By%2B3%7D%20%5Cright%29%20%5C%5C%5C%5C%5C%5C%20%5Clog%5Cleft%28%20%5Ccfrac%7Bx%2B4%7D%7B3%7D%20%5Cright%29%3D%28y%2B3%29%5Clog%5Cleft%28%202%20%5Cright%29%5Cimplies%20%5Ccfrac%7B%5Clog%5Cleft%28%20%5Cfrac%7Bx%2B4%7D%7B3%7D%20%5Cright%29%7D%7B%5Clog%282%29%7D%3Dy%2B3%20%5C%5C%5C%5C%5B-0.35em%5D%20%5Crule%7B34em%7D%7B0.25pt%7D%5C%5C%5C%5C%20~%5Chfill%20%5Ccfrac%7B%5Clog%5Cleft%28%20%5Cfrac%7Bx%2B4%7D%7B3%7D%20%5Cright%29%7D%7B%5Clog%282%29%7D-3%3D%5Cstackrel%7Bf%5E%7B-1%7D%28x%29%7D%7By%7D~%5Chfill)
Answer:
f-1(x) = log2 [ (x + 4)/3 ] - 3.
Step-by-step explanation:
Let y = 3(2^(x+3) - 4
3(2^(x + 3) = y + 4
2^(x + 3) = (y + 4)/3
x + 3 = log2 [(y + 4)/3]
x = log2 [ (y + 4)/3 ] - 3
Inverse f-1(x) = log2 [ (x + 4)/3 ] - 3
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