Logarithms are indices in disguise

$\log_a x = y \quad\text{means exactly}\quad a^y = x$

So a logarithm is an index (also called an exponent or power). $\log_2 8 = 3$ because $2^3 = 8$. Read it as "the power of 2 that gives 8".

Words:

• $a$ is the base (the number being raised to a power). It must be positive and not 1 (every power of 1 is 1, so base 1 could never give 8).
• $x$ is the argument (the number you take the log of).

Consequences worth memorising:

• $\log_a 1 = 0$ for every base $a$;
• $\log_a a = 1$;
• $\log_a x$ is only defined for $x > 0$, since $a^y$ is always positive.

Two bases have their own notation:

• $\lg x$ means $\log_{10} x$ (the log button);
• $\ln x$ means $\log_e x$ (the ln button), where $e \approx 2.718$ is a special number you'll meet in the next lesson.

The three laws (for positive $x$ and $y$, and any number $k$):

$\log_a (xy) = \log_a x + \log_a y$ $\log_a\!\left(\frac{x}{y}\right) = \log_a x - \log_a y$ $\log_a (x^k) = k\log_a x$

They are index laws in disguise. Try it with numbers: $\log_2 8 + \log_2 4 = 3 + 2 = 5$, and $\log_2 (8 \times 4) = \log_2 32 = 5$ ✓

That works because $8 \times 4 = 2^3 \times 2^2 = 2^{3+2}$: multiplying powers adds the exponents, and logs are exponents. (The full proof is in the deeper block below.)

The most useful form of the power law is backwards: $k\log_a x = \log_a(x^k)$.

That is how a number in front of a log is moved inside it. For example, $2\log_5 3 = \log_5 9$.