The exponent to which a base must be raised to produce a given number. If $b^a = c$, then $log_b(c) = a$.

The base of a common logarithm is 10. It is written as $log(c)$.

The base of a natural logarithm is $e$ (Euler's number, ≈ 2.71828). It is written as $ln(c)$.

The number you're taking the logarithm of.

The base is the number that is raised to a power to obtain the argument. It must be positive and not equal to 1.

A scale in which units represent a multiplicative change of the base, where each unit is a power of the base.

The exponent to which you raise the base to get the argument.

Euler's Number is the base of the natural logarithm, approximately equal to 2.71828.

The inverse function is $x = log_b(c)$

The argument of $log_b(c)$ is $c$.

Linear: Units are equally spaced, fixed increment. | Logarithmic: Units represent multiplicative change, power of the base.

$log(x)$: Base 10 | $ln(x)$: Base $e$ (Euler's number).

Exponential: $y = b^x$, rapid growth | Logarithmic: $y = log_b(x)$, slower growth, inverse of exponential.

$y = log_b(x)$: Vertical asymptote at $x = 0$, passes through (1, 0) | $y = b^x$: Horizontal asymptote at $y = 0$, passes through (0, 1)

$y = log_b(x)$: Domain is $(0, \infty)$ | $y = b^x$: Domain is $(-\infty, \infty)$

$y = log_b(x)$: Range is $(-\infty, \infty)$ | $y = b^x$: Range is $(0, \infty)$

$y = log_b(x)$: Approaches infinity at a decreasing rate | $y = b^x$: Approaches infinity at an increasing rate

$y = log(x)$: Derivative is $\frac{1}{x \cdot ln(10)}$ | $y = ln(x)$: Derivative is $\frac{1}{x}$

Linear: Suitable for data with evenly distributed values | Logarithmic: Suitable for data with values spanning several orders of magnitude

$y = log_b(x)$: Increasing function when $b > 1$, decreasing function when $0 < b < 1$ | $y = b^x$: Increasing function when $b > 1$, decreasing function when $0 < b < 1$

The domain is $x > 0$, meaning the graph exists only for positive values of $x$.

The range is all real numbers, meaning $y$ can take any real value.

If $b > 1$, the graph is increasing. If $0 < b < 1$, the graph is decreasing.

There is a vertical asymptote at $x = 0$.

The derivative of $y = e^x$ is also $e^x$, meaning the rate of change is always positive and equal to the function's value.

The derivative of $y = ln(x)$ is $1/x$, which is always positive for $x > 0$.

The derivative of $y = log(x)$ is $\frac{1}{x \cdot ln(10)}$, which is always positive for $x > 0$.

The x-intercept is always at $x = 1$, since $log_b(1) = 0$ for any valid base $b$.

The graph of $y = log_b(x)$ is concave down for $b > 1$ and concave up for $0 < b < 1$.

As $x$ approaches infinity, $y = log_b(x)$ also approaches infinity, but at a slower rate than a linear function.