The base ($a$) of the logarithm determines the shape and behavior of the graph $y = \log_a x$.
When the base $a$ is greater than 1, the function is increasing.
Behavior: The graph rises as $x$ increases.
Key Points: It always passes through (1, 0).
Example: The graph of $y = \log_2 x$.
$x=1 \implies y=0$
$x=2 \implies y=1$
$x=4 \implies y=2$
When the base $a$ is between 0 and 1, the function is decreasing.
Behavior: The graph falls as $x$ increases.
Key Points: It also always passes through (1, 0).
Example: The graph of $y = \log_{1/2} x$.
$x=1 \implies y=0$
$x=2 \implies y=-1$
$x=1/2 \implies y=1$
These are key rules for simplifying expressions and solving equations.
| Property | Rule | Description |
| Inverse Property 1 | $a^{\log_a b} = b$ | The base and $\log$ with the same base cancel out. |
| Equality | $\log_a m = \log_a n \implies m = n$ | If logs of the same base are equal, their arguments are equal. |
The direction of the inequality sign is critically dependent on the base $a$.
| Condition | Base a>1 | Base 0<a<1 |
| $\log_a x \ge 0$ | $\iff a>1, x \ge 1$ | $\iff 0<a<1, 0<x \le 1$ |
| $\log_a x \le 0$ | $\iff a>1, 0<x \le 1$ | $\iff 0<a<1, x \ge 1$ |
| $\log_a x < 0$ | $\iff a>1, 0<x < 1$ | $\iff 0<a<1, x > 1$ |
The rule for removing the logarithm depends entirely on the base:
When comparing two logarithms, the direction of the inequality for the arguments ($x$ and $y$) also depends on the base $a$: