1 Attachment(s)

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.

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 .

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

Re: Solving for x in logarithm problem

Re: Solving for x in logarithm problem

Quote:

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:

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

If so:

2. Use the base change formula:

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

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

3. Multiply through by ln(x):

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

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

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

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: $\displaystyle x = 4 ~\vee~x=\frac12 \cdot \sqrt[3]{4}$

Re: Solving for x in logarithm problem

thank you very much i got it