How to prove the logarithm product rule i.e. show that log_a(xy)=log_a(x)+log_a(y)
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Logarithms are mathematical functions that represent the exponent or index to which a base must be raised to obtain a given number. Logarithms have several laws that simplify the manipulation of these functions. Here are the fundamental logarithm laws:
1. *Product Rule:*
logₐ(xy) = logₐ(x) + logₐ(y)
This rule states that the logarithm of the product of two numbers is equal to the sum of the logarithms of the individual numbers.
2. *Quotient Rule:*
logₐ(x/y) = logₐ(x) - logₐ(y)
The quotient rule states that the logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator.
3. *Power Rule:*
logₐ(xⁿ) = n × logₐ(x)
This rule states that the logarithm of a number raised to a power is equal to the exponent times the logarithm of the base.
4. *Change of Base Formula:*
logₐ(x) = logₕ(x) / logₕ(a)
This formula allows you to change the base of a logarithm. Common choices for h are 10 (common logarithm) or e (natural logarithm).
These laws are useful for simplifying logarithmic expressions, solving equations involving logarithms, and performing various mathematical manipulations involving logarithmic functions.
