Questions: Exponential and Logarithmic Functions Converting between natural logarithmic and exponential equations Rewrite each equation as requested. (a) Rewrite as a logarithmic equation. e^x=7 (b) Rewrite as an exponential equation. ln 4=y

Solution Steps

To convert between exponential and logarithmic equations, we use the following relationships:

1. \( e^x = a \) can be rewritten as \( x = \ln(a) \).
2. \( \ln(a) = y \) can be rewritten as \( e^y = a \).

Let's apply these rules to the given equations.

(a) Rewrite \( e^x = 7 \) as a logarithmic equation:

• Use the relationship \( e^x = a \) can be rewritten as \( x = \ln(a) \).

(b) Rewrite \( \ln 4 = y \) as an exponential equation:

• Use the relationship \( \ln(a) = y \) can be rewritten as \( e^y = a \).

Given the equation \( e^x = 7 \), we can rewrite it in logarithmic form. Using the relationship \( e^x = a \) implies \( x = \ln(a) \), we have:

\[ x = \ln(7) \approx 1.9459 \]

For the equation \( \ln 4 = y \), we can convert it to exponential form. Using the relationship \( \ln(a) = y \) implies \( e^y = a \), we find:

\[ e^y = 4 \quad \text{where} \quad y = \ln(4) \approx 1.3863 \]

Final Answer