Solving Logarithmic Equations

A logarithm is the exponent to which a base must be raised to produce a given number. For example, log_b(a) = c means b^c = a.

The change of base formula states that log_b(a) = log_k(a) / log_k(b) for any positive k.

To solve log_x(1000) = 3, rewrite it in exponential form: x^3 = 1000. Therefore, x = 10.

log_b(m) + log_b(n). This property states that the logarithm of a product is the sum of the logarithms.

log_b(m) - log_b(n). This property states that the logarithm of a quotient is the difference of the logarithms.

n * log_b(m). This property states that the logarithm of a power is the exponent times the logarithm of the base.

Since the logs are equal, set the arguments equal: x^2 - 6 = 2x + 2. Rearranging gives x^2 - 2x - 8 = 0, which factors to (x - 4)(x + 2) = 0, so x = 4 or x = -2.

Rewrite in exponential form: 4x + 4 = 8^2 = 64. Thus, 4x = 60, so x = 15.

Combine the logs: log(x(x + 3)) = 1. Rewrite in exponential form: x(x + 3) = 10. This gives x^2 + 3x - 10 = 0, which factors to (x - 2)(x + 5) = 0, so x = 2 or x = -5.

The domain is x > 0, meaning the argument of the logarithm must be positive.