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Apply logarithm properties: Product, Quotient, Power, and Change of Base
logb(xy) = logb(x) + logb(y)
The log of a product equals the sum of the logs
logb(x/y) = logb(x) - logb(y)
The log of a quotient equals the difference of the logs
logb(xn) = n × logb(x)
The log of a power brings the exponent to the front
logb(x) = logc(x) / logc(b)
Convert between bases using division
Logarithm properties simplify complex expressions, solve equations, and convert between different forms. They're essential for calculus, solving exponential equations, and scientific calculations.
Use the product rule log(xy) = log(x) + log(y) when you have multiplication inside a logarithm and want to split it into a sum, or when combining sums back into a single log.
The power rule log(x^n) = n·log(x) brings exponents to the front as coefficients. This is incredibly useful for solving exponential equations and simplifying expressions with powers.
Most calculators only have ln and log₁₀ buttons. The change of base formula lets you calculate any logarithm using these available functions: log_b(x) = ln(x) / ln(b).
Yes! You can expand (split apart) or condense (combine) logarithms. For example, log(x) + log(y) = log(xy) condenses two logs into one, while log(xy) = log(x) + log(y) expands one log into two.
Common errors: log(x + y) ≠ log(x) + log(y) (no rule for addition inside log), log(x)·log(y) ≠ log(xy) (multiplication of logs isn't the same as log of a product), and forgetting that these rules only work with the same base.
Yes! These properties work with any valid logarithm base (positive, not equal to 1). Whether you use log₁₀, ln, log₂, or any other base, the properties apply the same way.
Remember that logs convert multiplication to addition (product rule), division to subtraction (quotient rule), and powers to multiplication (power rule). Logs "simplify" the operation to something easier.