Question

In Exercises \(33-36,\) solve the equation algebraically. Support your solution graphically. $$e^{0.05 t}=3$$

Short answer

The solution for \(t\) is approximately equal to \(t = ln(3) / 0.05\). This is also confirmed by the intersection point of the graphs of the functions \(y=e^{0.05t}\) and \(y=3\).

Step-by-step solution

Apply a natural logarithm on both sides

Since e is the base, recall that the natural logarithm(ln) is to the base e. Hence, apply ln on both sides to get rid of the e. This gives us \(ln(e^{0.05t}) = ln(3)\). According to the power rule of logarithms, the exponent can come down as a factor, hence our equation simplifies to \(0.05t * ln(e) = ln(3)\) or \(0.05t = ln(3)\) since ln(e)=1.

Solve for t

Next, isolate the variable t by dividing both sides of the equation by 0.05. Thus, we find that \(t = ln(3) / 0.05\). We can use a calculator to find the numerical value of \(t\).

Graphical Representation

Plot the graph of \(y=e^{0.05t}\) and \(y=3\) on the same axes. The value for t found above should be the x-coordinate of the intersection point of these two graphs. This ensures the validity of our solution.