Last updated on Oct 23, 2026

The Laws of Logs (Leaving Cert Mathematics): Flashcards

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The Laws of Logs

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Logarithms are the inverse of...

Exponentials

If $a^y = x$ , then $\log_a(x) = ?$

$y$

Constraints on base $a$ in $\log_a(x)$

$a > 0$ and $a \neq 1$

Constraint on argument $x$ in $\log_a(x)$

$x > 0$

Product Rule: $\log_a(xy) = ?$

$\log_a(x) + \log_a(y)$

Quotient Rule: $\log_a\left(\frac{x}{y}\right) = ?$

$\log_a(x) - \log_a(y)$

Power Rule: $\log_a(x^q) = ?$

$q \cdot \log_a(x)$

Negative Arg Rule: $\log_a\left(\frac{1}{x}\right) = ?$

$-\log_a(x)$

$\log_a(1) = ?$

$0$

Why does $\log_a(1) = 0$ ?

Because $a^0 = 1$ for any base $a$

Logarithm Identity: $a^{\log_a(x)} = ?$

$x$

Inverse Property: $\log_a(a^x) = ?$

$x$

Product Rule: log of product equals...

Sum of logarithms of factors

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