\( \begin{array}{ll}\text { 85. Express } \ln 2 x=4 \text { in exponential form } & \text { 24. Express } e^{5}=x \text { in logarithmic form } \end{array} \)

Sure, let's address both problems one by one. --- **Problem 85:** *Express \( \ln(2x) = 4 \) in exponential form.* The natural logarithm equation \( \ln(2x) = 4 \) can be rewritten in its exponential form by using the definition of the natural logarithm: \[ \ln(a) = b \quad \text{is equivalent to} \quad a = e^b \] Applying this to the given equation: \[ \ln(2x) = 4 \quad \Rightarrow \quad 2x = e^4 \] **Exponential Form:** \[ 2x = e^4 \] --- **Problem 24:** *Express \( e^5 = x \) in logarithmic form.* The exponential equation \( e^5 = x \) can be rewritten in logarithmic form using the definition of logarithms: \[ e^b = a \quad \text{is equivalent to} \quad \ln(a) = b \] Applying this to the given equation: \[ e^5 = x \quad \Rightarrow \quad \ln(x) = 5 \] **Logarithmic Form:** \[ \ln(x) = 5 \] --- **Summary:** 1. \( \ln(2x) = 4 \) in exponential form is \( 2x = e^4 \). 2. \( e^5 = x \) in logarithmic form is \( \ln(x) = 5 \).