Question

Solve for \(x\). $$ e^{x}=1 $$

Short answer

The solution to the given equation \(e^{x} = 1\) is \(x = 0\).

Step-by-step solution

Apply the Property of Logarithm

Start with the given equation \(e^{x} = 1\). The basic property of logarithm \(a^{b} = c\) can be written as \(b = \log_{a} c\). In this case, \(x\) refers to \(b\), \(e\) refers to \(a\), and 1 refers to \(c\). By applying this logarithmic property, we get \(x = \log_{e} 1\). The symbol for \(\log_{e}\) is often written as ln, so we can simplify this to \(x = \ln 1\).

Calculate the Natural Logarithm

Use the rule of natural logarithm, with any number 'a', \(\ln a = 0\) if \(a = 1\). Hence, \(\ln 1 = 0\). Replace \(\ln 1\) with 0 in the equation to solve for \(x\).

Solution

After simplification, we get \(x = 0\). Hence, the solution of \(e^{x} = 1\) is \(x = 0\).