Chapter 12/13.2

The properties of logarithms state that \( \log_a b + \log_a c = \log_a (bc) \) and \( \log_a b - \log_a c = \log_a \left( \frac{b}{c} \right) \).

The exponential form is \( 2^3=8 \).

A function is one-to-one if it never assigns the same value to two different domain elements, meaning if \( f(a) = f(b) \), then \( a = b \).

This can be expressed as \( 3(\log_a 6 - \log_a 2) = 3\log_a \left( \frac{6}{2} \right) = 3\log_a 3 = \log_a 27 \).

As \( x \) approaches infinity, \( f(x) \) approaches infinity as well, indicating an exponential growth.

The identity states that \( \log_a a = 1 \) for any base \( a \) where \( a > 0 \).

The change of base formula is given by \( \log_a b = \frac{\log_c b}{\log_c a} \) for any positive base \( c \).

It simplifies to \( \log_x c^4 + \log_x f^3 - \log_x k^8 = 4\log_x c + 3\log_x f - 8\log_x k \).

A constant function is one that does not change its value; for example, \( h(x) = 2 \) is constant and is not one-to-one.

Logarithms are the inverse operations of exponentiation; for example, if \( b^y = x \), then \( \log_b x = y \).