Solving Exponential Equations

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

To solve an exponential equation with the same base, set the exponents equal to each other. For example, if $$2^x = 2^3$$ , then $$x = 3$$ .

The natural logarithm, denoted as $$\text{ln}$$ , is the logarithm to the base $$e$$ , where $$e$$ is approximately equal to 2.718.

You can use logarithms to solve exponential equations by taking the logarithm of both sides. For example, for $$5^{2x} = 50$$ , take $$\text{ln}$$ of both sides: $$\text{ln}(5^{2x}) = \text{ln}(50)$$ .

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

To round a number to the nearest thousandth, look at the digit in the fourth decimal place. If it is 5 or greater, round up the third decimal place by one.

The value of $$e^0$$ is 1.

A function is exponential if it can be expressed in the form $$f(x) = a \times b^x$$ , where $$a$$ is a constant and $$b$$ is a positive real number.

To find the value of an exponential function at a given point, substitute the value of $$x$$ into the function. For example, for $$A(t) = 20,000(0.88)^t$$ , substitute $$t$$ to find the value.

The base of an exponential function determines the rate of growth or decay. A base greater than 1 indicates growth, while a base between 0 and 1 indicates decay.