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Part 1 of 1
Logarithm problems require applying standard identities to combine or decompose expressions. The primary identities are: log_b(x * y) = log_b(x) + log_b(y), log_b(x / y) = log_b(x) - log_b(y), and log_b(x^k) = k * log_b(x).
The change of base formula allows conversion between bases: log_b(x) = ln(x) / ln(b). Always state the initial domain restriction before simplifying logarithmic equations.
Worked example
Solve log_2(x) + log_2(x - 2) = 3. Domain: x > 2. Combine logs: log_2(x(x - 2)) = 3 implies x(x - 2) = 2^3 = 8. Rearrange: x^2 - 2x - 8 = 0 implies (x - 4)(x + 2) = 0. Solutions: x = 4 (valid) or x = -2 (rejected by domain). Answer: x = 4.
1.The expression log(A) + log(B) is equal to log(A + B).
2.The equation log_3(x^2) = 2 log_3(x) holds for all real x != 0.
3.The value of log_5(125) is equal to 3.
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