# Thread: Solving for x in logarithm problem

1. ## Solving for x in logarithm problem

The equation (attached as image) has
(a)one irrational solution
(b)no prime solution
(c)two real solutions
(d)one integral solution

i would like to get help on how to find the possible values of x

i solved the equation by applying the base changing property and then doing
L.C.M of both equations but after that i am unable to solve further?

Any help will be highly appreciated.

2. ## Re: Solving for x in logarithm problem

Show what you tried so far.

What results?

I suggest: Do change of base with base of 2 .

3. ## Re: Solving for x in logarithm problem

2/log2(x) + 6/ (log2(x)+1)=3
i got this result
after that what should i do

4. ## Re: Solving for x in logarithm problem

Solve for log_2 (x) .

5. ## Re: Solving for x in logarithm problem

Originally Posted by sumedh
The equation (attached as image) has
(a)one irrational solution
(b)no prime solution
(c)two real solutions
(d)one integral solution

i would like to get help on how to find the possible values of x

i solved the equation by applying the base changing property and then doing
L.C.M of both equations but after that i am unable to solve further?

Any help will be highly appreciated.
I assume that the question reads:

$\log_{x^2}(16)+\log_{2x}(64)=3$

If so:

2. Use the base change formula:

$\dfrac{4\ln(2)}{\ln(x^2)}+\dfrac{6\ln(2)}{\ln(2x)} =3$

$\dfrac{4\ln(2)}{2\ln(x)}+\dfrac{6\ln(2)}{(\ln(2)+\ ln(x))}=3$

3. Multiply through by ln(x):

$2\ln(2)+\dfrac{6\ln(2) \ln(x)}{(\ln(2)+\ln(x))}=3 \ln(x)$

Multiply through by (ln(2)+ln(x)):

$2 \ln(2) (\ln(2)+\ln(x))+6\ln(2) \ln(x)=3 \ln(x)(\ln(2)+\ln(x))$

4. Expand the brackests and collect like terms. You'll get a quadratic equation in ln(x). Solve for ln(x) and consequently for x.

5. For confirmation only: There are 2 solutions: $x = 4 ~\vee~x=\frac12 \cdot \sqrt[3]{4}$

6. ## Re: Solving for x in logarithm problem

thank you very much i got it