Question

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

(a) \({\bf{ln}}\left( {{\bf{4}}x + {\bf{2}}} \right) = {\bf{3}}\) (b) \({e^{{\bf{2}}x - {\bf{3}}}} = {\bf{12}}\)

Short answer

(a) The exact value is \(x = \frac{1}{4}\left( {{e^3} - 2} \right)\), and approximated value is 4.521.

(b) The exact value is \(x = \frac{1}{2}\left( {3 + \ln 12} \right)\), and approximated 2.742.

Step-by-step solution

Find the solution for part (a)

Solve the equation \(\ln \left( {4x + 2} \right) = 3\) by using logarithmic properties is shown below:

\(\begin{aligned}\ln \left( {4x + 2} \right) &= 3\\{e^{\ln \left( {4x + 2} \right)}} &= {e^3}\\4x + 2 &= {e^3}\\x &= \frac{1}{4}\left( {{e^3} - 2} \right)\\ \approx 4.521\end{aligned}\)

Thus, \(\ln \left( {4x + 2} \right) \approx 4.521\).

Find the solution for part (b)

Solve the equation \({e^{2x - 3}} = 12\) by using the logarithmic properties as shown below:

\(\begin{aligned}{e^{2x - 3}} &= 12\\\ln \left( {{e^{2x - 3}}} \right) &= \ln 12\\2x - 3 &= \ln 12\\x &= \frac{1}{2}\left( {3 + \ln 12} \right)\\ \approx 2.742\end{aligned}\)

Thus, \({e^{2x - 3}} \approx 2.742\).