Solve for x if 5+lnx=14/lnx.
substitute y=ln(x) ....(1)
to get
5+y=14/y
Solve the resulting quadratic equation in y.
Note that the domain of ln(x) is ℝ+\0, reject the value of y≤0.
Solve for x from (1) above.
ok, so i got e^-7 and e^2
does that mean i reject e^-7 because its less than or equal to 0?
Your answers are correct,
x=e-7 or x=e².
In fact, the value of y in y=ln(x) can be anything in ℝ, so your answers are correct. The restriction for non-negative values are on x only. My apologies.
To solve the equation 5 + ln(x) = 14/ln(x) for x, we can follow these steps:
Step 1: Get rid of the fractions
To eliminate the fraction, we can multiply both sides of the equation by lnx. This will result in:
ln(x) * (5 + ln(x)) = ln(x) * (14/ln(x))
Step 2: Expand and simplify
Using the distributive property, the equation becomes:
5ln(x) + (ln(x))^2 = 14
Step 3: Rearrange the equation
Rearranging the equation to have the quadratic term on the left side and the constant on the right side gives us:
(ln(x))^2 + 5ln(x) - 14 = 0
Step 4: Solve the quadratic equation
To solve the quadratic equation, let's substitute ln(x) with a variable, let's say u:
u^2 + 5u - 14 = 0
Now we can use factoring, completing the square, or the quadratic formula to solve for u. Alternatively, we can use a graphing calculator to find the roots of the quadratic equation.
Step 5: Substitute back and solve for x
Once we find the solutions for the variable u, we substitute them back into ln(x) to find the values for x. Remember that x must be greater than zero since we took the natural logarithm.
After solving, we will find the values of x that satisfy the equation 5 + ln(x) = 14/ln(x).