log(2X + 1) - log(X + 2)= logX

Sure, let's solve the equation step by step.

The equation is: log(2X + 1) - log(X + 2) = logX

Step 1: Use the logarithmic property log(a) - log(b) = log(a/b) to combine the left side of the equation.

log((2X + 1) / (X + 2)) = logX

Step 2: If logA = logB, then A = B. So, we can remove the logarithms from both sides.

(2X + 1) / (X + 2) = X

Step 3: Cross-multiply to get rid of the fraction.

2X + 1 = X * (X + 2)

Step 4: Distribute the X on the right side.

2X + 1 = X^2 + 2X

Step 5: Rearrange the equation to set it equal to zero (this is a standard form for quadratic equations).

0 = X^2 + 2X - 2X - 1

0 = X^2 - 1

Step 6: Solve for X using the square root property.

X = sqrt(1) or X = -sqrt(1)

So, the solutions are X = 1 and X = -1. However, we must check these solutions in the original equation because logarithms are undefined for negative numbers and zero.

Checking X = 1:

log(2*1 + 1) - log(1 + 2) = log1 log3 - log3 = 0 0 = 0

Checking X = -1:

log(2*-1 + 1) - log(-1 + 2) = log(-1) log(-1) - log(1) = log(-1)

This is undefined because we can't take the logarithm of a negative number. So, the only solution to the equation is X = 1.