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Break apart logarithmic expressions using product, quotient, and power rules
log(x²y³) = 2log(x) + 3log(y)
log(x⁴/y²) = 4log(x) - 2log(y)
log(x²y/z³) = 2log(x) + log(y) - 3log(z)
log((x³)²) = 6log(x)
Expanding a logarithm means breaking it apart into simpler pieces using logarithm properties. For example, log(xy) expands to log(x) + log(y). This makes complex expressions easier to work with.
Expand logarithms when solving equations, taking derivatives, or when you need to evaluate each component separately. It's the opposite of condensing, which combines multiple logs into one.
Three key rules: Product rule (log(xy) = log(x) + log(y)), Quotient rule (log(x/y) = log(x) - log(y)), and Power rule (log(x^n) = n·log(x)). Apply these systematically from outside to inside.
No! There's no logarithm property for addition inside the log. log(x + y) cannot be simplified to log(x) + log(y). The product rule only works for multiplication, not addition.
A coefficient in front becomes an exponent inside: 3log(x) = log(x³). When expanding, exponents come out as coefficients. This power rule works in both directions.
Remember that √x = x^(1/2). So log(√x) = log(x^(1/2)) = (1/2)log(x). Roots become fractional exponents that you can bring to the front using the power rule.
No, these expansion rules work for any valid logarithm base. Whether you're using log₁₀, ln, log₂, or any other base, the product, quotient, and power rules apply exactly the same way.
Plug in specific numerical values for the variables and calculate both the original and expanded forms. They should give the same result. Also, try condensing your expansion back to see if you get the original expression.