Video: Solving Logarithmic Equations Involving Exponents

Solve ln 9^(7𝑥 − 9) = ln 74 for 𝑥, giving your answer to the nearest hundredth.

02:16

Video Transcript

Solve ln nine to the power of seven 𝑥 minus nine equals ln 74 for 𝑥, giving your answer to the nearest hundredth.

So, the first thing we can see in this question that we’re dealing with ln which is the natural logarithm. So therefore, what we need to do is rewrite our equation using a natural log law. And the particular natural log law that we’re gonna use is the power law. And that tells us that if we have ln or the natural logarithm of 𝑥 to the power of 𝑦, this is equal to 𝑦 multiplied by ln 𝑥. Well, we can apply this to the left-hand side of our equation. And when we do, we’re gonna get seven 𝑥 minus nine multiplied by ln nine is equal to ln 74.

So then, what we’re gonna have is seven 𝑥 minus nine is equal to ln 74 divided by ln nine. We get that because we’re gonna divide each side of the equation by ln nine. And that’s to leave the 𝑥 term on its own on the left-hand side. And then, if we add nine to each side of the equation, then we will be left with the 𝑥 term on its own on the left-hand side, which is what we’re looking for as we said. So, we’re gonna get seven 𝑥 is equal to ln 74 over ln nine plus nine.

So then finally, because we want single 𝑥, what we’re gonna do is divide both sides of the equation by seven. And when we do that, we get 𝑥 is equal to ln 74 over ln nine plus nine. And then, this is all divided by seven. And when we do this, we’re gonna get the answer of 𝑥 is equal to 1.56555 continuing. And that’s the answer we get for our 𝑥. But have we finished here?

Well, no because if we check the question, it wants it to the nearest hundredth. So, we’re gonna need to round. So therefore, if we look at the second decimal place, because this is the hundredths, we can see that we’ve got six where the number after it is a five. So therefore, we’re gonna round the six up to a seven. So therefore, we can say the final answer is 𝑥 is equal to 1.57, and that’s to two decimal places.

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