19. Find the inverse of $f(x)=\log (x+3)$.

Step 1 ：Given the function \(f(x)=\log (x+3)\), we first replace \(f(x)\) with \(y\) to get \(y=\log (x+3)\).

Step 2 ：Next, we switch the roles of y and x to get \(x=\log (y+3)\).

Step 3 ：To solve for y, we need to get rid of the logarithm. We do this by using the property of logarithms that states \(a=\log_b c\) is equivalent to \(b^a=c\).

Step 4 ：So, we rewrite \(x=\log (y+3)\) as \(10^x=y+3\) (since the base of the logarithm is 10).

Step 5 ：Finally, we solve for y to get \(y=10^x-3\).

Step 6 ：So, the inverse of \(f(x)=\log (x+3)\) is \(f^{-1}(x)=10^x-3\).

Step 7 ：To check our work, we can substitute \(f^{-1}(x)\) into the original function and see if we get x back.

Step 8 ：Substituting \(f^{-1}(x)\) into \(f(x)\) gives us \(f(f^{-1}(x))=\log ((10^x-3)+3)=\log (10^x)=x\), which confirms that \(f^{-1}(x)=10^x-3\) is indeed the inverse of \(f(x)=\log (x+3)\).

Step 9 ：\(\boxed{f^{-1}(x)=10^x-3}\) is the final answer.