Solving Exponential Equations

An exponential equation is an equation in which a variable appears in the exponent. For example, $$2^x = 8$$ .

To solve $$12^{2x-10} = 1$$ , recognize that $$1 = 12^0$$ . Therefore, set the exponents equal: $$2x - 10 = 0$$ , leading to $$x = 5$$ .

The domain of the function $$f(x) = 12^x$$ is all real numbers, represented as $$(-\text{∞}, \text{∞})$$ .

Take the logarithm of both sides: $$x + 1 = \frac{\text{log}(3)}{\text{log}(8)}$$ , leading to $$x = -0.472$$ .

Take the logarithm of both sides: $$x = \frac{\text{log}(80)}{\text{log}(6)}$$ , which approximates to $$2.4457$$ .

The exponential form is $$x^3 = 25$$ .

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

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

The logarithmic form is $$\text{log}_b a = x$$ .

The value is 2, since $$10^2 = 100$$ .