Video Transcript
Given that log to the base 45 of 46 is equal to 𝑎 and log of 5.86 is equal to 𝑏, find the value of six raised to the power 𝑎 times 𝑏 plus one. Give your answer correct to four decimal places.
We’re given two logarithmic equations. That’s log to the base 45 of 46 is equal to 𝑎 and log of 5.86 is equal to 𝑏. And we’re asked to find the value of six raised to the power 𝑎 times 𝑏 plus one. But before we start trying to calculate anything, there are a couple of things to note.
The first is that while both 𝑎 and 𝑏 are logarithmic expressions, it’s important to note that the logarithms have different bases. 𝑎 is the logarithm to the base 45 of 46, while for 𝑏 the base is not stated. And since this is written log as opposed to ln, or the natural logarithm, which is the logarithm to the base 𝑒, where 𝑒 is Euler’s number, we assume that 𝑏 is the logarithm to the base 10 of 5.86. So let’s write this in. And we have logarithm to base 10 of 5.86 is equal to 𝑏.
The second thing to note is that we can use one of the laws of exponents to simplify the expression six raised to the power 𝑎 times 𝑏 plus one. Recalling that 𝑃 raised to the power 𝑞 plus 𝑟 is 𝑃 raised to the power 𝑞 multiplied by 𝑃 raised to the power 𝑟, we can rewrite six raised to the power 𝑎 times 𝑏 plus one as six raised to the power 𝑎 times 𝑏 multiplied by six raised to the power one. And that’s where 𝑞 is equal to 𝑎 times 𝑏 and 𝑟 is equal to one. And since anything to the power one is itself, we can rearrange this to six multiplied by six raised to the power 𝑎 times 𝑏. This gives us perhaps a little less room for error when we want to input 𝑎 and 𝑏, our logarithmic expressions, into our calculators.
Now, as things stand, there isn’t a great deal we can do to simplify anything further. We might think we could use the fact that the logarithm to the base 𝑃 of 𝑚 is equal to 𝑛 is equivalent to 𝑃 raised to the power 𝑛 is equal to 𝑚. But applying this to our equation for 𝑎, that’s the logarithm to the base 45 of 46 is equal to 𝑎, this simply gives us 45 raised to the power 𝑎 is 46. This means that the 𝑎th root of 46 is equal to 45.
Similarly, applying this to our equation for 𝑏, we have 10 raised to the power 𝑏 is equal to 5.86. And this means that 10 is the 𝑏th root of 5.86. This is all very interesting, but it does not help us to calculate six times six raised to the power 𝑎𝑏. That’s six raised to the power 𝑎𝑏 plus one. So this is no help at all.
We also don’t have any sums, products, or powers in our logarithms. So the usual laws of logarithms don’t help either. And it seems with the information we have, all we can do is try and work out the value of six times six raised to the power 𝑎𝑏 using our calculators.
Now, if we’re lucky, our calculators will let us specify the base of 45. So we can calculate the logarithm to the base 45 of 46, that’s 𝑎, directly. In fact, evaluating this in our calculators gives us 1.0057738. And if we calculate 𝑏 using the logarithm, we have 0.7678976 and so on. For accuracy though, into our calculators we put six multiplied by six raised to the power log to the base 45 of 46 multiplied by log to the base 10 of 5.86. This gives us 23.94097425 and so on, which to four decimal places is 23.9410.
We can finish by noting that if our calculator doesn’t let us specify the base of the logarithm, we can use the fact that the logarithm to the base 45 of 46 is the logarithm to the base 10 of 46 divided by the logarithm to the base 10 of 45. And here we’re using the change of base formula, which is covered in another lesson. Our calculation then becomes six multiplied by six raised to the power log to the base 10 of 46 divided by log to the base 10 of 45 times the log to the base 10 of 5.86. And this of course gives us the same answer six raised to the power 𝑎 times 𝑏 plus one is equal to 23.9410 to four decimal places.
