Question

express the equation in logarithmic form.
(a) \\( e^{x + 1} = 0.5 \\)

Explanation:

Step1: Recall the exponential - logarithmic relation

We know that for the exponential equation \(y = e^{x}\), the corresponding logarithmic form is \(\ln(y)=x\) (since the natural logarithm \(\ln\) is the logarithm with base \(e\), i.e., \(\log_{e}(y)=x\) is written as \(\ln(y) = x\)).

Step2: Apply the relation to the given equation

Given the equation \(e^{x + 1}=0.5\). Let \(y = 0.5\) and the exponent be \(x+1\) in the exponential form \(y=e^{k}\) (where \(k=x + 1\)).
Using the relation \(\ln(y)=k\) (where \(y = e^{k}\)), we substitute \(y = 0.5\) and \(k=x + 1\) into the logarithmic form.
So we get \(\ln(0.5)=x + 1\)

Answer:

\(\ln(0.5)=x + 1\)