Question Video: Computing Logarithms by Using Laws of Logarithms Mathematics

Without using a calculator, find the approximate value of log₃ 5/243, given that log₃ 5 ≈ 1.465.

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Video Transcript

Without using a calculator, find the approximate value of log to base three of five over 243, given that log to base three of five is approximately equal to 1.465.

So straightaway, what we can do is take a look at log to the base three of five over 243. And what we can do is we can think about rewriting this in another form. And we can do that using one of our log laws. Well, what the log law that we’re looking at tells us is that log to base 𝑎 of 𝑥 over 𝑦 is equal to log to the base 𝑎 of 𝑥 minus log to base 𝑎 of 𝑦. You can see that the base is staying the same throughout.

Well, we can see that, in our expression, we’ve got log to base three of five over 243. So therefore, if we’re using this log law, we can think about 𝑎 being three, our 𝑥 five, and our 𝑦 243. So now what we can do is apply that law. So therefore, what we can rewrite our expression as is log to base three of five minus log to base three of 243. But we might think about how is this any more useful.

Well, for a start, we know what log to base three of five is approximately equal to. So that’s 1.465. However, if we look at log to base three of 243, how are we gonna work out the value of this? Well, first of all, we’d think, well, we know how to rewrite 243 as three to the power of something. And that’s because we know that three to the power of five is equal to 243. So we can write it in exponent form.

Okay, so now we’ve rewritten. So we’ve got log to base three of five, and we know that we know the approximate value of that, minus log to base three of three to the power of five. So what we do now is bring in another one of our log laws into play. So this log law is that log to base 𝑎 of 𝑥 to the power of 𝑛 is equal to 𝑛 multiplied by log to base 𝑎 of 𝑥. So again, we can identify our parts: our 𝑎, 𝑥, and 𝑛.

So now what we’re gonna do is rewrite it once more. So now our expression is log to base three of five minus five log to base three of three. So it’s now that we can do something, because now what we can do is deal with the right-hand term and turn it into a value. And that’s because there’s something special about log to base three of three.

So now we can use a relationship from our log laws, and that is that log to base 𝑎 of 𝑎 is equal to one. So it doesn’t matter what the value of 𝑎 is. We’ll always get one, which just means that our right-hand term is gonna be five multiplied by one, which is equal to five. So now if we substitute in our approximate value for log to base three of five, and that’s 1.465, we can say that the expression is approximately equal to 1.465 minus five. So what we do when we calculate this is we get an approximate value of log to base three of five over 243 of negative 3.535.

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