3 lnx - 4lnx^4 = 2

Question

Solve for x and give answers as exact solutions

Answer 1

First, this problem involves natural logarithms, which have base e.  The exact solution will be in terms of e .

We can begin by taking advantage of the logarithmic property of exponents, i.e., ln x²  = 2 lin x, ln x⁴ = 4 ln x, etc.

Thus, 3 ln x - 4 ln x⁴ = 2  can be rewritten as:
          3 ln x - 4(4) ln x = 2 , or

          3 ln x - 16 ln x  = 2, which is

                  -13 ln x  = 2, or 

                    ln x  = -2/13

The base is e, so we convert the ln function into an exponential function:

                    x = e ^-2/13 

This is the exact solution. The decimal solution to 4 places, is x = 0.8574

Answer 2

Jack's solution is perfect. Another way to get there, though is to think about it this way:

3 ln x = ln(x^3) and 4 ln (x^4) = ln (x^4)^4 = ln (x^16)

Use the rule that ln a - ln b = ln (a/b) to get:  

ln (x^3) - ln(x^16) = ln (x^3/x^16)

ln (x^3/x^16) = ln (x^-13) = -13 ln x

The rest is the same. -13 ln x = 2, ln x = -2/13, x = e^(-2/13)

 

Answer 3

3ln(x) - 4lin(x⁴) = 2

3ln(x) - 4*4ln(x) = 2

3ln(x) - 16ln(x) = 2

-13ln(x) = 2

ln(x) = -2/13

x = e^-2/13

(This is approximately .8574039192)