Apply logarithm laws to expand complex expressions into sums and differences.
Last updated: March 2026 | By ForgeCalc Engineering
Expanding logarithms is the process of using the laws of logarithms to break down a single logarithmic expression containing products, quotients, or powers into multiple simpler logarithmic terms.
This is a crucial skill in algebra and calculus, as it allows you to simplify complex equations, solve for variables in exponents, and differentiate logarithmic functions more easily.
Expand log(x²y / z³):
1. Apply Quotient Rule: log(x²y) - log(z³)
2. Apply Product Rule: log(x²) + log(y) - log(z³)
3. Apply Power Rule: 2log(x) + log(y) - 3log(z)
Final Answer: 2log(x) + log(y) - 3log(z)
No. There is no law for the logarithm of a sum. You can only expand products, quotients, and powers.
ln stands for the natural logarithm, which is a logarithm with base e (Euler's number, ≈ 2.718).
No. All expanded terms must keep the same base as the original expression.
Condensing is the opposite of expanding. It involves taking multiple log terms and combining them into a single log expression.
The product rule states: log_b(xy) = log_b(x) + log_b(y). When you have a product inside a logarithm, you split it into a sum of logarithms.
The quotient rule states: log_b(x/y) = log_b(x) - log_b(y). When you have a fraction inside a logarithm, you can split it into a difference of logarithms.
The power rule states: log_b(x^n) = n·log_b(x). An exponent inside the logarithm can be brought out front as a coefficient.
Yes, as long as all logarithms in the expression have the same base. If bases differ, you would need to convert them first using the change of base formula.
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