Solve for x: 3logx-2logx=2logx+1-3logx-1

Step-by-Step Solution

To solve the equation 3^(log x) - 2^(log x) = 2^(log x + 1) - 3^(log x - 1):

1. Simplify using Substitution: Let log x = m. The equation becomes: 3^m - 2^m = 2^(m + 1) - 3^(m - 1)

2. Rearrange the terms: Group the terms with base 3 on one side and base 2 on the other: 3^m + 3^(m - 1) = 2^(m + 1) + 2^m

3. Factorize: Use the property a^(m - 1) = a^m / a: 3^m + (3^m / 3) = (2^m * 2) + 2^m Factor out 3^m on the left and 2^m on the right: 3^m * (1 + 1/3) = 2^m * (2 + 1) 3^m * (4/3) = 2^m * 3

4. Solve for m: Rearrange to isolate (3/2)^m: 3^m / 2^m = 3 / (4/3) (3/2)^m = 3 * (3/4) = 9/4 (3/2)^m = (3/2)^2 Therefore, m = 2.

5. Find x: Since log x = m and m = 2, then log_10 x = 2. x = 10^2 = 100.