Exponential Equations Review

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

Convert $$\frac{1}{125}$$ to an exponential form: $$125 = 5^3$$ , so $$\frac{1}{125} = 5^{-3}$$ . Therefore, set the exponents equal: $$a + 2 = -3$$ , leading to $$a = -5$$ .

A function exhibits growth if its value increases as the input increases. In exponential functions, if the base is greater than 1, it represents growth.

A function exhibits decay if its value decreases as the input increases. In exponential functions, if the base is between 0 and 1, it represents decay.

Since the base (1.35) is greater than 1, the function is a growth function.

The general form of an exponential function is $$y = ab^x$$ , where $$a$$ is a constant, $$b$$ is the base, and $$x$$ is the exponent.

Since the bases are the same, set the exponents equal: $$3 = 4x + 2$$ . Solving gives $$x = \frac{1}{4}$$ .

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

Convert 16 to an exponential form: $$16 = 2^4$$ . Therefore, set the exponents equal: $$x = 4$$ .

To graph an exponential function, plot points for various values of $$x$$ , noting that the function approaches the x-axis but never touches it (asymptote).