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.