Converting between natural logarithmic and exponential equations
Rewrite each equation as requested.
(a) Rewrite as an exponential equation.
\[
\ln x=6
\]
(b) Rewrite as a logarithmic equation.
\[
e^{9}=y
\]
(a) $\square$
(b)
Answer
Final Answer: \n(a) The exponential form of the equation \(\ln x = 6\) is \(e^6 = x\), which gives \(x \approx \boxed{403.43}\). \n(b) The logarithmic form of the equation \(e^9 = y\) is \(\ln y = 9\), which gives \(y \approx \boxed{8103.08}\).
Step 1 ：Given the equation \(\ln x = 6\), we can rewrite it in exponential form as \(e^6 = x\).
Step 2 ：Calculating \(e^6\), we find that \(x \approx 403.43\).
Step 3 ：Given the equation \(e^9 = y\), we can rewrite it in logarithmic form as \(\ln y = 9\).
Step 4 ：Calculating \(e^9\), we find that \(y \approx 8103.08\).
Step 5 ：Final Answer: \n(a) The exponential form of the equation \(\ln x = 6\) is \(e^6 = x\), which gives \(x \approx \boxed{403.43}\). \n(b) The logarithmic form of the equation \(e^9 = y\) is \(\ln y = 9\), which gives \(y \approx \boxed{8103.08}\).
