The properties of logarithms are essential tools for simplifying expressions and solving logarithmic equations. They stem from the definition of a logarithm and its relationship with exponents. Here are the primary properties:

1. Product Rule

$\log_b(M \cdot N) = \log_b(M) + \log_b(N)$
This property shows that the logarithm of a product is the sum of the logarithms of the factors.

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2. Quotient Rule

$\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$
This property shows that the logarithm of a quotient is the difference of the logarithms of the numerator and denominator.

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3. Power Rule

$\log_b(M^p) = p \cdot \log_b(M)$
This property states that the logarithm of a power is the exponent multiplied by the logarithm of the base.

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4. Change of Base Formula

$\log_b(M) = \frac{\log_c(M)}{\log_c(b)}$
This allows changing the base of a logarithm, useful for calculation or converting to a base (like 10 or $e$) supported by calculators.

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5. Logarithm of 1

$\log_b(1) = 0$
The logarithm of 1 in any base is 0, since $b^0 = 1$.

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6. Logarithm of the Base

$\log_b(b) = 1$
The logarithm of a base to itself is always 1, since $b^1 = b$.

Would you like to see examples, derivations, or applications of these properties?

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5 Related Questions:

1. How do these properties help simplify logarithmic equations?
2. What are some real-life applications of logarithms in science or engineering?
3. How does the change of base formula work for $\ln$ (natural logarithms)?
4. Can these rules be used to expand and condense logarithmic expressions?
5. How do you graph logarithmic functions using these properties?

Tip:

Always check the domain of the argument $M$ or $N$ in logarithmic expressions—they must be positive real numbers!