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Combine multiple logarithms into a single logarithmic expression
2log(x) + 3log(y) = log(x²) + log(y³) = log(x²y³)
4log(x) - 2log(y) = log(x⁴) - log(y²) = log(x⁴/y²)
2log(x) + log(y) - 3log(z) = log(x²y/z³)
log(x) + log(y) = log(xy)
Condensing logarithms means combining multiple log terms into a single logarithm. For example, log(x) + log(y) condenses to log(xy). It's the opposite of expanding logarithms.
Condense when solving logarithmic equations, simplifying expressions for evaluation, or when you need a single log term instead of multiple terms. It's especially useful before applying the definition of logarithms.
Three key rules: log(x) + log(y) = log(xy) (addition becomes multiplication), log(x) - log(y) = log(x/y) (subtraction becomes division), and n·log(x) = log(x^n) (coefficient becomes exponent).
Coefficients become exponents first. Convert 3log(x) to log(x³) before combining with other terms. This uses the power rule in reverse: n·log(x) = log(x^n).
Addition creates products in the numerator, subtraction creates products in the denominator. For example: log(x) + log(y) - log(z) = log(xy/z).
Only logarithms with the same base can be condensed together. You cannot combine log₁₀(x) + ln(y) directly; they must first be converted to the same base using change of base formula.
Expanding is the opposite of condensing. While condensing combines multiple logs into one, expanding breaks one log into multiple terms. Both use the same logarithm properties, just in reverse.
Expand your condensed form and see if you get back the original expression. Or plug in numerical values for variables and verify that both forms give the same result.