Exponential and Logarithm Review

A vertical asymptote is a line x = a where a function approaches infinity or negative infinity as the input approaches a. It indicates that the function is undefined at that point.

A horizontal asymptote is a line y = b that a function approaches as the input approaches positive or negative infinity. It indicates the behavior of the function at extreme values.

The domain is all real numbers except x = 0, which can be expressed as D: (-∞, 0) ∪ (0, ∞).

The range is all real numbers, which can be expressed as R: (-∞, ∞).

Set the argument of the logarithm to zero: x^2 = 0. The vertical asymptote is at x = 0.

The formula for exponential decay is $$P(t) = P_0 e^{-kt}$$ , where P_0 is the initial amount, k is the decay constant, and t is time.

Set up the equation as $$146 = 700 e^{-0.00513t}$$ and solve for t.

The value is 2, since 5^2 = 25.

This uses the property $$\text{log}_a b - \text{log}_a c = \text{log}_a \frac{b}{c}$$ .

The result is -1, since $$\frac{125}{625} = \frac{1}{5}$$ and $$\text{log}_5 \frac{1}{5} = -1$$ .