Could someone please help with logarithms? ?

Question

What is the value of x in the equation:

log 24 - log 3 = log x

2 2 5

The numbers on the bottom are the bases.

Answer 1

x = 125

The log of a factorable value (like 24:  2*12, 3*8, 4*6) equals the sum of the logs of each factor. Another way we can say that is:  log xy = log x + log y. Let's use in reverse first (log x + log y = log xy) and apply it to the subtraction on the left side:

[log(2) 24] - [log(2) 3] = log(2) (24/3) = log(2) 8

After all, of added logs give you the log of the values multiplied by each other, then log x - log y must equal log x/y, right. Right.

Now, since we have different bases on each side, we can move forward a couple ways. We could convert the logs to common bases, base 5 say, but that's gonna look like a nightmare:

[log(5) 8] / [log(5) 2] = log(5) x

and an easier way is available. So let's try that easier way:

log(2) 8 = log(2) 2^3

and we know that log x^y = y * log x, so:

log(2) 8 = 3 * log(2) 2

and log(2) 2 must be 1 because what power do you raise anything to to get that value? x^1 = x and log(x) x = 1.

So, we know the left side equals 3. Now things look easy!

log(5) x = 3

so 5^3 = x = 125