Question Video: Solving Base e Exponential Equations

Video Transcript

Solve three 𝑒 to the 𝑥 power minus four equals eight, giving your answer correct to three decimal places.

We have the equation three times 𝑒 to the 𝑥 power minus four equals eight. Solving this equation means finding the value for 𝑥 that makes this statement true. And to do that, we’ll need to isolate 𝑥.

The first thing we can do is move the four to the other side by adding four to both sides. On the left side, minus four plus four cancels each other out. And we’re left with three times 𝑒 to the 𝑥 power. And on the right side, eight plus four equals 12.

We’re still trying to isolate 𝑥, but we haven’t done that yet. We can divide both sides of the equation by three, leaving us with 𝑒 to the 𝑥 power on the left and 12 divided by three equals four on the right side. At this point, it seems like there’s nothing else we can do. However, we can take the natural log of both sides of the equation. And because of the log rules, we know the natural log of 𝑒 to the 𝑥 power is the same thing as 𝑥 times the natural log of 𝑒.

And so what we can say is that 𝑥 times the natural log of 𝑒 is equal to the natural log of four. And we know the value of the natural log of 𝑒 is one. And so we’re saying 𝑥 times one, which means that, on the left side of our equation, we now only have 𝑥. And that 𝑥 value is equal to the natural log of four. And we wanna round the natural log of four to three decimal places.

If you put the natural log of four into your calculator, it will tell you that that is equal to 1.38629436 continuing. To round to three decimal places, we’ll consider the fourth decimal place, where we see a two, which means we should round down to 1.386. We could also plug in three times 𝑒 to the 1.386 power minus four into our calculator and check that we get eight. When you do this, you get 7.99, which is almost eight. And that verifies our answer. 𝑥 is equal to 1.386.