Unit 5 Review

A logarithmic function is the inverse of an exponential function. It can be expressed as $$y = \text{log}_b(x)$$ , where $$b$$ is the base, and it answers the question: 'To what exponent must $$b$$ be raised to produce $$x$$ ?'.

An exponential function is a mathematical function of the form $$f(x) = b^x$$ , where $$b$$ is a positive constant called the base. It describes growth or decay processes.

Exponential functions and logarithmic functions are inverses of each other. If $$y = b^x$$ , then $$x = \text{log}_b(y)$$ .

A logarithmic function will show a linear increase in output as the input increases exponentially.

The base of a logarithm is the number that is raised to a power to obtain a given number. For example, in $$\text{log}_b(x)$$ , $$b$$ is the base.

The domain of a logarithmic function is all positive real numbers, i.e., $$x > 0$$ .

The range of a logarithmic function is all real numbers, i.e., $$y \text{ can be any real number}$$ .

First, find $$f(-1) = 2(-1) = -2$$ . Then, substitute into $$g(x)$$ : $$g(-2) = 2(-2)^2 - 1 = 8 - 1 = 7$$ .

The graph of a logarithmic function increases slowly and approaches the y-axis but never touches it (asymptote at x=0). It passes through the point (1,0).

Linear growth increases by a constant amount, while exponential growth increases by a constant percentage of the current value.