Question Video: Solving Base e Exponential Equations Mathematics

Solve 2𝑒^(3𝑥) + 11 = 18, giving your answer correct to three decimal places.

02:19

Video Transcript

Solve two 𝑒 to the power of three 𝑥 plus 11 equals 18, giving your answer correct to three decimal places.

So now in order to solve this equation, we’re gonna start in the same way we’d solve any equation. So what we want to do is have the terms involving 𝑥 on one side. So what we’re gonna do is keep the two 𝑒 to the power of three 𝑥 on the left-hand side. So the first step that we’re gonna do is we’re going to subtract 11 from both sides of the equation. And once we’ve done that, what we have is two 𝑒 to the power of three 𝑥 equals seven. So great. Now what’s the next step?

Well, now what we want to do is divide through by two. That’s because we’ve got two 𝑒 to the power of three 𝑥. So we’re gonna divide through by two. So now what we have is 𝑒 raised to the power three 𝑥 is equal to 3.5. And we got that because seven divided by two is 3.5. So great, we’re now at the stage where we have to do something a little bit different.

Well, the next step we’re gonna need to do is we’re gonna take the natural logarithm of both sides of the equation. And we’re gonna do that because of a relationship we know. And that is that the natural logarithm of 𝑒 raised to the power of 𝑥 is equal to 𝑥.

So therefore, if we take the natural algorithm of both sides of the equation, we’re gonna have the natural algorithm of 𝑒 raised to the power of three 𝑥 is equal to the natural algorithm of 3.5. So then, if we utilise the relationship we’re talking about, we’re gonna get three 𝑥 is equal to the natural algorithm of 3.5. So when we divide each side of the equation by three, what we’re gonna get is 𝑥 is equal to the natural algorithm of 3.5 over three. So this is gonna give us 𝑥 is equal to 0.41758765 continuous. So have we finished here?

Well, no, because if we take a look at the question, it asked us to give the answer correct to three decimal places. So therefore, what we do is we take a look at the third decimal place, which in this case is seven. And then we look at the number after it. This is the deciding number. Well, we can see that this is five. So because it is five or above, we’re gonna round the seven to an eight.

So therefore, we can say the solution to the question is 𝑥 is equal to 0.418. And that’s to three decimal places.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.