Difficult Log Problems

Question

1.)   (log base3 r)^2 + 8 = log base3 r^-6

    Solve for r

 

2.)   3(log base5 y)^2  = log base5 y^15

    Solve for y

 

I have tried many different things to solve these two problems, but cannot come up with the correct solution.

Can anyone help guide me in the right direction?

 

Thank you

Answer 1

Here's help on the first one:

Let u = log₃r  i.e. log, base 3, of r.

 

Then rewrite the equation as u² + 8 = -6u  and the solution to this quadratic, by factoring, is u = -2 or -4.

 

If log, base 3, of r = -2, exponentiating with base 3 tells you r = 3^-2 = 1/9.

If log, base 3, of r = -4, exponentiating with base 3 tells you r = 3^-4 = 1/81.

Answer 2

These are both looking like they'll end up as quadratic equations.

 

If you bring the -6 out front of the log and rearrange a little you can get the first equation to become

 

(log₃r)² + 6log₃r + 8 = 0   which is like x² + 6x + 8 = 0

 

Think you can take it from there?

 

The second one is pretty similar.  Bring down the power, move it to one side, factor and solve.