Calculating logarithms

Japanese school year: Math II

What you learn

Learn the formal definition of logarithms as the inverse of exponentiation, along with operational rules and the change-of-base formula. Logarithms are widely used to handle wide-ranging scales such as sound intensity, acidity, and data entropy. Prior proficiency with exponent properties is essential for understanding logarithmic transformations.

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Key points

This formula breaks the logarithm of a product $MN$ into the sum of separate logarithms (where $a > 0, a \neq 1$ , and $M, N > 0$ ). It simplifies multiplication by converting it into addition.

\log_a(MN) = \log_a M + \log_a N

This formula expresses the logarithm of a quotient $\frac{M}{N}$ as the difference of two logarithms. It is used to simplify fractional logarithms or combine logarithmic terms.

\log_a\left(\frac{M}{N}\right) = \log_a M - \log_a N

This formula brings an exponent $k$ on the argument $M$ to the front as a coefficient. It is used to simplify powers inside logarithms or pull down unknown exponents in equations.

\log_a M^k = k\log_a M

This base-change formula converts a logarithm from base $a$ to a new base $c$ (where $c > 0, c \neq 1$ ). It is essential when matching different bases in logarithmic expressions.

\log_a b = \frac{\log_c b}{\log_c a}

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Practice 1Coming soon