Logarithmic and Exponential Equations

A logarithm is the exponent to which a base must be raised to produce a given number. For example, if $$b^y = x$$ , then $$\text{log}_b(x) = y$$ .

The base of the common logarithm is 10.

Exponential functions can be expressed as logarithmic functions and vice versa. If $$y = b^x$$ , then $$x = \text{log}_b(y)$$ .

$$n = \frac{1}{2}$$ .

$$\text{log}_{5}25 = 2$$ because $$5^2 = 25$$ .

The change of base formula states that $$\text{log}_b(a) = \frac{\text{log}_k(a)}{\text{log}_k(b)}$$ for any base $$k$$ .

$$x = \text{log}_{5}(17)$$ .

$$\text{log}_{10}(100) = 2$$ because $$10^2 = 100$$ .

$$\text{log}_{2}(8) = 3$$ because $$2^3 = 8$$ .

An exponential function is a mathematical function of the form $$f(x) = a \times b^x$$ , where $$a$$ is a constant, $$b$$ is the base, and $$x$$ is the exponent.