I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 1.9 : Exponential And Logarithm Equations
1. Find all the solutions to \(12 - 4{{\bf{e}}^{7 + 3\,x}} = 7\). If there are no solutions clearly explain why.
Show All Steps Hide All Steps
Start SolutionThere isn’t all that much to do here for this equation. First, we need to isolate the exponential on one side by itself with a coefficient of one.
\[ - 4{{\bf{e}}^{7 + 3\,x}} = - 5\hspace{0.5in} \Rightarrow \hspace{0.5in}{{\bf{e}}^{7 + 3\,x}} = \frac{5}{4}\] Show Step 2Now all we need to do is take the natural logarithm of both sides and then solve for \(x\).
\[\begin{align*}\ln \left( {{{\bf{e}}^{7 + 3\,x}}} \right) & = \ln \left( {\frac{5}{4}} \right)\\ 7 + 3x & = \ln \left( {\frac{5}{4}} \right)\\ x & = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{1}{3}\left( {\ln \left( {\frac{5}{4}} \right) - 7} \right) = - 2.25895}}\end{align*}\]Depending upon your preferences either the exact or decimal solution can be used.