Video: Finding the Solution Set of an Exponential Equation

Video Transcript

Find to the nearest hundredth the value of 𝑥 for which two to the power of 𝑥 plus eight equals nine.

Now the first stage to actually solve this equation is take logs of both sides and this gives us log of two to the power of 𝑥 plus eight equals log of nine. And what we say here by log is log to the base 10. So if we don’t write a base, it automatically means log to the base 10. So if you actually press log in your calculator, then it’s actually gonna calculate as if it’s log to the base 10.

Okay, great, but what do we do now? Well, we’re actually gonna use a log law to help us solve the equation. And the one we’re gonna use is that log 𝐴 to the power of 𝑛 is equal to 𝑛 log 𝐴. And by applying this law, we get 𝑥 plus eight log two equals log nine. And we get this because 𝑥 plus eight was our 𝑛 and two was our 𝐴.

Okay, great, so now what’s the next step? Well, the next step is to actually divide each side by log two, which gives us 𝑥 plus eight is equal to log nine over log two. So then, we actually subtract eight from each side. So therefore, we can say that 𝑥 is equal to log nine over log two minus eight, which if we put into calculator, we get 𝑥 is equal to negative 4.8300749986.

So then, we take a look back at the question to see how it wants us to actually leave our answer. And the question says that it wants us to leave it to the nearest hundredth. So therefore, we know that 𝑥 is equal to negative 4.83.