Question Video: Solving Exponential Equations

Video Transcript

Find, to the nearest thousandth, the value of 𝑥 such that 𝑒 to the power of four 𝑥 minus three equals 19.

Let’s just remind ourselves about this number 𝑒 first of all. 𝑒 is an irrational number, which means that it can’t be expressed as a fraction 𝑎 over 𝑏. 𝑒 is equal to this decimal 2.718281828 continuing. Now, it looks here as if the pattern of digits after the decimal point repeats because we have one eight, two eight, one eight, two eight. But if we were to go to a higher number of decimal places, we would see that the digits in these places do actually change from this pattern.

Now, in the equation we’ve been given, we have this number 𝑒. And then it’s been raised to the power of four 𝑥 minus three to give the result 19. So, in order to solve this equation, we need to use the inverse operation, or inverse process, of raising something to a power. And that inverse process is called a logarithm.

A logarithm is, therefore, an alternative way of describing a power relationship. If we have the relationship 𝑎 to the power of 𝑐 is equal to 𝑏, this can be written equivalently as the logarithm to the base 𝑎 of 𝑏 is equal to 𝑐. So the value 𝑎 here is the base. That’s the number that is originally being raised to the power. The value 𝑐 is the power. And the value 𝑏 is the result. It’s the value that we’d get when we raise that value 𝑎 to the power of 𝑐.

So our first step in solving this equation is going to be to take a logarithm of each side. And as the base in the equation is 𝑒, we’re going to be using logarithms with a base of 𝑒. So to take logarithms of each side, we can just write log with a base of 𝑒 before each of the two sides of this equation. Giving log to the base 𝑒 of 𝑒 to the power four 𝑥 minus three is equal to log to the base 𝑒 of 19. Now a logarithm with a base 𝑒 is called the natural logarithm. And it’s often abbreviated to ln or ln. So we can say that ln of 𝑒 to the power of four 𝑥 minus three is equal to ln of 19.

Now, let’s think about how we can simplify this. When we take a logarithm, we’re asking ourselves a question. On the left-hand side, 𝑒 will be the base and 𝑒 to the power of four 𝑥 minus three is the result. So the question we’re asking ourselves is, “What power do we have to raise the base 𝑒 to in order to get 𝑒 to the power of four 𝑥 minus three?”

Well, the answer to that question is just four 𝑥 minus three. If we raise 𝑒 to the power of four 𝑥 minus three, then we’ll get 𝑒 to the power of four 𝑥 minus three. And this should also be intuitive, because raising 𝑒 to a power and then taking a natural logarithm of this are inverse operations. So we’re just left with the power itself. We can’t simplify the right hand-side nicely, so we’ll just leave it as ln of 19. We now have four 𝑥 minus three equals ln of 19.

We need to solve this equation for 𝑥, which is on the left-hand side of this equation. We have four 𝑥 minus three. So there are two more steps that we need to go through. First, we need to add three to each side of this equation, giving four 𝑥 equals three plus ln of 19. Now, I’ve written it that way round deliberately because we want to make it clear that it’s ln of 19 only. If I’d written ln 19 plus three, then there’s a risk that we may think it’s ln of 19 plus three rather than ln of 19 plus three. If I write the three first, then there’s no chance of us making this mistake.

The next step is to divide both sides of the equation by four, giving 𝑥 equals three plus ln of 19 all over four. Remember, ln of 19 is just a number. It’s the answer to the question, “What power do I have to raise 𝑒 to to get 19?” So it just acts like a constant when we’re solving this equation.

If we were asked to give an exact answer, we could leave our answer in this format. However, we’ve been asked to give our answer to the nearest thousandth. So we now need to evaluate using our calculator. Your calculator will have an ln button, which will just look like this. Or it may have a button like this, log and then blanks, where the base and the result are. If you use this second button, then you can specify what base you want to use for your logarithm. In this case, we need to specify base 𝑒.

Typing this fraction into our calculator then gives 1.4861097 continuing. To round to the nearest thousandth — that’s the third decimal place — we need to consider the value in the fourth decimal place, which is a one. And as this is less than five, we’re rounding down, giving 1.486. So we found that, to the nearest thousandth, the value of 𝑥 such that 𝑒 to the power of four 𝑥 minus three equals 19 is 1.486.

We can, of course, check our answer by substituting this value of 1.486 back into the left-hand side of the equation, giving 𝑒 to the power of four multiplied by 1.486 minus three. And evaluating this on a calculator gives 18.99166 continuing, not exactly 19 because we’ve rounded our answer, but close enough. So we can be confident that our answer of 1.486 is correct.