Question

Solve the inequality for \(x\). $$ 1<\ln x<100 $$

Short answer

The solution to the inequality is \(e < x < e^{100}\).

Step-by-step solution

Solve the Left Side of Inequality

Start by isolating the inequality \(\ln{x} > 1\). Apply the rule of exponential to transform the logarithm into a more workable expression. This leads to \(x > e^1 = e\).

Solve the Right Side of Inequality

Next, solve the inequality \(\ln{x} < 100\). Using the same rule of exponential, we have \(x < e^{100}\).

Combine the Results

The solutions of the left and right inequalities both apply, therefore the overall solution to the inequality is \(e < x < e^{100}\).