Question

61-62: Solve each inequality for \(x\).

61. (a) \(\ln x < 0\) (b) \({e^x} > 5\)

Short answer

(a) The solution of inequality is \(0 < x < 1\).

(b) The solution of inequality is \(x > \ln 5\).

Step-by-step solution

Solve the inequality for \(x\)

a)

Solve the inequality for \(x\) as shown below:

\(\begin{aligned}\ln x < 0\\x < {e^0}\\x < 1\end{aligned}\)

The solution of the original inequality is \(0 < x < 1\) because the domain of the function \(f\left( x \right) = \ln x\) is \(x > 0\).

Thus, the solution of inequality is \(0 < x < 1\).

Solve the inequality for \(x\)

Solve the inequality for \(x\) as shown below:

\({e^x} > 5\)

Take natural logarithms on both sides of inequality as shown below:

\(\begin{aligned}\ln {e^x} < \ln 5\\x > \ln 5\end{aligned}\)

Thus, the solution of inequality is \(x > \ln 5\).