Solve the equation for x: In(x)+In(x-1)=1

Question

Please help I'm so confused

Answer 1

To solve you must use the properties of adding logarithmic functions.

 

Step 1: Rewrite the left side of the equation

 

ln(x)+ln(x-1) = ln(x)(x-1)

 

Step 2: New equation is now

 

ln(x)(x-1)=1

 

Step 3: Rewrite the equation where each side is now the exponent and the base is e

 

e^ln(x)(x-1) = e¹

 

Step 4: Simplify the equation (remember e raised to the ln function cancels each other out)

 

x(x-1)=e¹

x²-x-e=0

 

Step 5: Solve the quadratic equation x²-x-e=0

 

Because we cannot factor with the "e" value we must use the quadratic formula

 

a=1

b= -1

c= -e

 

x=-b+- sqrt(b²-4ac)/2a

 

x=-(-1)+-sqrt((-1)²-4(1)(-e))/2(1)

 

x=1 +-sqrt((1+4e))/2

 

x= (1 +- sqrt(11.87312731))/2

 

x= 2.2228735 or x= -1.2228735  <-- cannot calculate a negative x value

 

Step 6: Plug in the x value above to initial equation to check answer

 

In( 2.2228735)+In( 2.2228735-1)=1

 

1=1 CORRECT!!