$log_b(c) = a$

$b^a = c$

$M = log(I/S)$, where M is magnitude, I is intensity, and S is the intensity of a standard earthquake.

If $b^x = y$, then $x = log_b(y)$.

For $log_b(x) = y$, $b$ is the base, $x$ is the argument, and $y$ is the exponent.

$log_a(b) = \frac{log_c(b)}{log_c(a)}$

$log(x) = log_{10}(x)$

$ln(x) = log_e(x)$

$b^{log_b(x)} = x$ and $log_b(b^x) = x$

$log_b(xy) = log_b(x) + log_b(y)$

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 $0 < b < 1$

Logs 'undo' exponents. If $b^x = c$, then $x = log_b(c)$. They essentially reverse each other's operations.

Units are equally spaced, each representing a fixed increment.

They are perfect for displaying data that spans many orders of magnitude, compressing large ranges and making trends easier to see.

A logarithmic scale represents values using the logarithm of the quantity. Equal distances on the scale represent equal ratios.

Exponential form expresses a number as a base raised to a power, while logarithmic form expresses the power to which the base must be raised to obtain the number.

If the base were 1, $1^x$ would always be 1, regardless of the value of $x$. This would make the logarithm undefined for any number other than 1.

The natural logarithm is significant because it is based on the number $e$, which appears frequently in calculus and other areas of mathematics. It simplifies many calculations and is used in various models of natural phenomena.

Because logarithms are the inverse of exponential functions, and exponential functions with real bases always produce positive values. Therefore, the logarithm of a non-positive number is undefined in the real number system.

By using logarithms, large values are transformed into smaller values, making it easier to visualize and analyze data that spans several orders of magnitude.

As $x$ approaches 0, the value of $y = log_b(x)$ approaches negative infinity, assuming $b > 1$. This indicates that the function has a vertical asymptote at $x = 0$.