Question

Solve for \(x\) accurate to three decimal places. (a) \(\ln x=2\) (b) \(e^{x}=4\)

Short answer

The approximate solutions are \(x = 7.389\) for \(\ln x = 2\) and \(x = 1.386\) for \(e^{x} = 4\).

Step-by-step solution

Solve \(\ln x = 2\)

The equation here is based on the natural logarithm. To solve \( \ln x = 2 \), the inverse of the natural log is the exponentiation of base \(e\). Therefore, we can write \(x = e^{2}\). Calculate this using the math value of \(e = 2.71828\) to get the accurate approximation of \(x\).

Solve \(e^{x} = 4\)

This equation uses the natural exponentiation. To solve \(e^{x} = 4\), we can use the natural logarithm (since it's the inverse of the exponentiation). Hence, \(x = ln(4)\). Calculate this to get the accurate approximation of \(x\).