Question

57-60 Solve each equation for x. Give both an exact value and decimal approximation, correct to three decimal places.

(a) \({\bf{ln}}\left( {{\bf{ln}}x} \right) = {\bf{0}}\) (b) \(\frac{{{\bf{60}}}}{{{\bf{1}} + {e^{ - x}}}} = {\bf{4}}\)

Short answer

(a) The exact value is \(x = e\). The approximated value is 2.718.

(b) The exact value is \(x = - \ln 14\). The approximated value is \( - 2.639\).

Step-by-step solution

Find the solution for part (a)

Solve the equation \(\ln \left( {\ln x} \right) = 0\) by using the logarithmic properties as shown below:

\(\begin{aligned}\ln \left( {\ln x} \right) &= 0\\{e^{\ln \left( {\ln x} \right)}} &= {e^0}\\\ln x &= 1\\x &= e\\x \approx 2.718\end{aligned}\)

Thus, the value is \(x = e \approx 2.718\).

Find the solution for part (b) 

Solve the equation \(\frac{{60}}{{1 + {e^{ - x}}}} = 4\) by using the logarithmic properties as shown below:

\(\begin{aligned}\frac{{60}}{{1 + {e^{ - x}}}} &= 4\\1 + {e^{ - x}} &= 15\\{e^{ - x}} &= 14\\\ln {e^{ - x}} &= \ln 14\\x &= - \ln 14\\x \approx - 2.639\end{aligned}\)

Thus, the value is \(x = - \ln 14 \approx - 2.639\).