Logarithmic functions are the inverses of exponential functions. They 'undo' each other.

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

Because the inverse exponential function has a horizontal asymptote at y = 0, restricting the domain of the logarithm to positive values.

As x approaches infinity, y also approaches either positive or negative infinity, depending on the base and any transformations.

A horizontal shift changes the vertical asymptote and, consequently, the domain. For example, $\log_b(x+k)$ shifts the asymptote to x = -k, and the domain becomes x > -k.

Because they are always either increasing or decreasing and always concave up or concave down.

The argument of a logarithm must be positive because you cannot raise a base to any power and get a non-positive result.

It reflects the graph across the x-axis, changing increasing functions to decreasing and vice versa, and affecting the end behavior.

Vertical shifts and stretches can change the range but since the range is all real numbers, only vertical shifts change the graph.

If output values are proportional over equal input intervals, then the function is logarithmic.

Exponential: Rapid increase, unbounded | Logarithmic: Slow increase, bounded by asymptote.

$\log_b(x)$: Increasing (if b > 1), positive y-values for x > 1 | $-\log_b(x)$: Decreasing (if b > 1), negative y-values for x > 1.

Horizontal: Changes the domain and asymptote | Vertical: Changes the range (though range is all real numbers).

$\log(x*y)$: Can be expanded to $\log(x) + \log(y)$ | $\log(x+y)$: Cannot be simplified further.

b > 1: Increasing function | 0 < b < 1: Decreasing function.

Exponential: Domain is all real numbers, range is y > 0 | Logarithmic: Domain is x > 0, range is all real numbers.

$\log_b(x)$: Vertical asymptote at x = 0 | $b^x$: Horizontal asymptote at y = 0. They are reflections across y = x.

Logarithmic: Often involves combining logs and converting to exponential form | Exponential: Often involves isolating the exponential term and taking the logarithm of both sides.

Vertical Stretch: Makes the graph steeper | Vertical Compression: Makes the graph less steep.

Product Rule: $\log_b(MN) = \log_b(M) + \log_b(N)$ | Quotient Rule: $\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$

$y = \log_b(x)$

$\lim_{x \to 0^+} a\log_b(x) = \pm \infty$

$\lim_{x \to \infty} a\log_b(x) = \pm \infty$

$g(x) = \log_b(x + k)$

$b^y = x$

$\log_b(1) = 0$

$\log_b(b) = 1$

$\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$

$\log_b(MN) = \log_b(M) + \log_b(N)$

$\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$