Logarithm Rules and Formulas List
List of logarithm formulas:
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Logarithm definition: logb^x = y if b^y = x, where x > 0, b > 0 and b ≠ 1.
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Common logarithm: log x = log10^x. The common logarithm has a base of 10.
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Natural logarithm: ln x = loge^x. The natural logarithm has a base of e, which is approximately 2.71828.
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Product rule: logb (xy) = logb x + logb y. The logarithm of the product of two numbers is equal to the sum of the logarithms of each individual number.
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Quotient rule: logb (x/y) = logb x - logb y. The logarithm of the quotient of two numbers is equal to the difference of the logarithms of each individual number.
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Power rule: logb (x^n) = n logb x. The logarithm of a number raised to a power is equal to the product of that power and the logarithm of the base number.
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Change of base formula: logb x = logc x / logc b. This formula is used to change the base of a logarithm from one base to another.
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Logarithmic identity: logb 1 = 0. The logarithm of 1 to any base is always equal to 0.
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Inverse of logarithmic function: b^{logb^x} = x. The exponential function with base b and the logarithmic function with base b are inverse functions of each other.
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Logarithmic differentiation rule: If f(x) = u(x)^v(x), then log e f(x) = v(x) log e u(x) Differentiating both sides with respect to x gives,
f’(x)/f(x) = v’(x)log e u(x) + v(x)(d/dx)log e u(x)
- Change of base formula for natural logarithm:
ln a = log e^a. This formula is used to change the base of a natural logarithm from e to another base.
These are some of the most commonly used logarithm formulas, and they are used in a wide variety of mathematical applications