Subject: having trouble with a problem
Name: Beth
Who are you: Teacher

Having trouble solving and explaining the following problem to my group of Algebra 2 students.

any help would be appreciated on how to solve without using the change of base formula for logarithms in the solution and check of the solution!!!

log256 (x) + log16 (x) + log4 (x) + log 2 (x) = 7/4

Thanks

We have two responses for you.

Hello Beth.

Let's say y = log4(x). Then 4^y = x. In other words, x = 2^2y.

Now take the log in base 2 of each side:
log₂(x) = log₂ (2^2y) = 2y (log₂ (2) ) = 2y = 2 log₄(x).

So you can substitute log₄ (x) with (1/2) log₂ (x).

Repeat as needed to solve for x.

Hope this helps,
Stephen La Rocque.

and

Beth,

You need to start by getting rid of the logs.

You have

256^{log256 (x) + log16 (x) + log4 (x) + log 2 (x)} = 256^7/4.

Simplify the right hand side by first taking a 4th root. As for the left hand side, since the bases involved are powers of 2, you can break it into a product of 4 parts, the first of which is
256^log256(x) = x, the next is 256^log16(x) = 16^2log16(x) = x² and so on. You should end up with x¹³ and then you can solve for x.

Penny