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\[f(x) = \log_{3}(\sqrt(x) + 3 \ \]

Log base 3 square root x plus 3

How do you find the inverse of this/

start with the fact that \(\log_3(\sqrt{x})=\frac{1}{2}\log_3(x)\)

is it the log of the whole thing?

No just log base 3 square root x

i mean is it
\[f(x)=\log_3(\sqrt{x}+3)\] or
\[f(x)=\log_3(\sqrt{x})+3\]

2nd one

ok then start with
\[f(x)=\frac{1}{2}\log_3(x)+3\]

you need the steps?

Yes please . I always get confused when there's stuff like log base 3

1) subtract 3
2) multiply by 2
3) raise 3 to everything

\[x=\frac{1}{2}\log_3(y)+3\]
\[x-3=\frac{1}{2}\log_3(y)\]
\[2x-6=\log_3(y)\]
\[3^{2x-6}=y\]

last step because
\[\log_b(y)=x\iff b^x=y\]

Ohhh!!! Thank you so much !

yw