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Logarithm Rules & Properties
Algebra โข Grades 10, 11, 12
Algebra
Logarithm Rules & Properties
All essential logarithm properties: Product, Quotient, Power, and Change of Base.
The Formula
log(xy) = log(x) + log(y)
Used for: Simplifying logarithmic expressions, solving exponential equations, and changing bases
bBase of the logarithm (b > 0, b โ 1)
x, yPositive arguments (x > 0, y > 0)
kPower / exponent coefficient
โกLive Interactive Quick Solver
Instant Calculationlog_10(100):2.0000
Change of Base: ln(100) / ln(10) = 4.6052 / 2.3026 = 2.0000
\log_b(xy) = \log_b x + \log_b y, \quad \log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y, \quad \log_b(x^k) = k \log_b x
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๐ฏ Key Concepts & Rules
Essential principles for calculating with this formula:
๐ชต
Product: log(xy) = log(x) + log(y)Logarithm of product is sum of logarithms.
โ
Quotient: log(x/y) = log(x) - log(y)Logarithm of quotient is difference of logarithms.
โก
Power: log(xแต) = k ยท log(x)Bring the exponent to the front as a multiplier.
๐ Worked Examples & Step-by-Step Solutions
Example 1: Simplify logโ(8) + logโ(4)
Problem:
Simplify logโ(8) + logโ(4).1
Product Rule
logโ(8 ร 4) = logโ(32)2
Calculate
2โต = 32 โ logโ(32) = 5Answer: 5
๐ก Pro Tips & Common Mistakes
Change of Base
log_b(a) = ln(a) / ln(b) = log(a) / log(b).
log(1) = 0 and log_b(b) = 1
Fundamental logarithm identities.
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