Find the logarithm of 1728 to the base of 2×√(3

Question

Find the logarithm of 1728 to the base of 2×√(3) without a calculator. Show working out

Answer 1

Hello Josh!

I hope I can help you today!

This problem is actually a lot less difficult than it seems at first, mostly because the answer is a nice whole number.

I'll do my best to explain my recommended approach to this problem.

First, I just wanted to mention that there are tutorials out there on how to do this by hand, but they are somewhat tricky to follow. That being said, sometimes it is nice to use a calculator to just get an idea of how hard this problem will be.

Entering in log(1728)/log(2 * sqrt(3)) yields 6.

(Most calculators don't have a logarithm button that allows you to change the base, so you can use the formula:

log base a of b = log (b) / log (a) instead to enter them into the calculator)

This tells us that solving this by hand should be relatively easy, since we get the whole number 6.

That aside, let's begin tackling the problem!

So, we want to calculate:

log_2×√31728

Working with something like 2 * sqrt(3) is really not fun, so let's suck the 2 into the radical

2*√3 = √(4 * 3)

(in order to bring it inside the radical you have to "undo" the square root function, so you just square it)

= √12

So we are working with base √(12).

log_2×√31728 = n

log_√121728 = n

Let's see if we can't get rid of the radical too.

You know how you can square both sides of an equation?

Here, we are going to use both sides of the equation as an exponent, like so:

√12^{log√12 (1728)} = (√12)ⁿ

On the left side the base of the logarithm is the same, so it cancels out:

1728 = (√12)ⁿ = (12^1/2)ⁿ=12^n/2

Notice that we turned the square root into a power of 1/2, so we could combine it with the exponent of n.

1728 = 12^n/2

Squaring both sides will get rid of the square root, but squaring 1728 by hand sounds like no fun.

So instead, let's put 1728 into it's prime factorization.

It is even (ends in 8), so we can divide by 2.

Keep doing this until you can divide by 2 no longer, and you get:

27 * 2⁶ = 12^n/2

27 is 3³, so we end up with:

3³×2⁶ = 12^n/2

Squaring both sides yields:

(3³×2⁶)² = 12ⁿ

3⁶×2¹² = 12ⁿ

Now, there are 2 ways of solving it from here, but they are essentially the same when you get down to it.

The first way is to take log base 12 of both sides.

The other way is to keep it in the current format of exponents.

Let's keep it as it is, since this is easier.

If we were to factor the right side, we would get:

12ⁿ=(3x4)ⁿ

We know that 4 = 2², so the right side is implying that we can manipulate the left side to get the two sides in terms of 3 and 4, and perhaps even 12.

2¹²=(2²)⁶

Thus we have the left side equaling:

3⁶×2¹²=3⁶×(2²)⁶ = (3x2²)⁶ = 12ⁿ

and taking it one step further:

(3x2²)⁶ = (3x4)⁶ = 12⁶ = 12ⁿ

(12)⁶ = 12ⁿ

Therefore n = 6.

I am not sure if this is what your teacher was looking for, but this is one way to solve the problem "by hand".

I hope this is interesting and helpful, even if it wasn't exactly what the teacher was looking for.

Please let me know if you have any questions, or want me to explain anything or want to know about alternative methods for finding the logarithm when it is not a whole number.