Questions: Rewrite each equation as requested. (a) Rewrite as an exponential equation. ln x=2 (b) Rewrite as a logarithmic equation. e^y=4

Solution Steps

To rewrite the given equations, we need to understand the relationship between logarithmic and exponential forms. For part (a), we convert the natural logarithm equation to its equivalent exponential form. For part (b), we convert the exponential equation to its equivalent logarithmic form.

We start with the equation given in part (a): \[ \ln x = 2 \] To convert this to its exponential form, we use the property that if \(\ln a = b\), then \(a = e^b\). Thus, we rewrite the equation as: \[ x = e^2 \] Calculating \(e^2\) gives us approximately: \[ x \approx 7.3891 \]

Next, we consider the equation from part (b): \[ e^y = 4 \] To convert this to its logarithmic form, we use the property that if \(a^b = c\), then \(\log_a c = b\). Therefore, we rewrite the equation as: \[ y = \ln 4 \] Calculating \(\ln 4\) gives us approximately: \[ y \approx 1.3863 \]

Final Answer