Next Generation Advanced Algebra and Functions Study Guide for the ACCUPLACER Test

Exponential and Logarithmic Equations

An exponential equation is an equation that contains a variable as an exponent: \(3^x = 9\), for example.

Exponential equations are used to model physical phenomena like radioactive decay, bacterial growth, and computer processing power.

A logarithmic equation is an equation that contains a logarithmic expression: \(log(4x) + log(3 + \sqrt{x}) = 8\), for example.

Logarithmic equations are used to model physical phenomena like the strength of earthquakes, the intensity of sounds, and the strength of acids.

Exponentials and logarithms exhibit an inverse relationship; and, just as multiplication can be used to “undo” the division operation, a logarithm can be used to “undo” an exponential; likewise, an exponential can be used to “undo” a logarithm

Let’s consider the exponential equation: \(4^{5x - 2} = 64\).

Begin by rewriting \(64\) in terms of \(4\) raised to an exponent:

\(4^{5x - 2} = 4^3\), because the same base is on both sides, the exponents can be equated:

\(5x - 2 = 3\), so \(x = 1\)

Now a logarithmic equation example: Solve \(6 + 3\ln{\frac{x}{9} - 4} = 3\)

Begin by isolating the natural logarithm:

\[\ln{\frac{x}{9} - 4} = -1\]

To undo the natural logarithm function, use \(e\) as the base, and exponentiate both sides of the equation:

\(e^{\ln{\frac{x}{9} - 4}} = e^{-1}\), which gives

\(\frac{x}{9} - 4 = \frac{1}{e}\), and isolating \(x\):

\[x = \frac{9}{e} + 36 \approx 39.3\]

Graphing

The general form of an exponential equation is: \(y = a^x\), where \(a\) is a constant. When \(a > 1\), the curve represents exponential growth. When \(1 > a > 0\), the curve represents exponential decay.

An exponential function is vertically translated by adding or subtracting a constant from the general form: \(y = a^x + c\), where \(a\) and \(c\) are constants.

An exponential function is horizontally translated by adding or subtracting a constant from the exponent: \(y = a^{x + b}\), where \(a\) and \(b\) are constants.

Let’s look at an example: \(y = (\frac{1}{4})^{x - 2} + 3\)

It can be determined that the function represents exponential decay and that the graph will be horizontally translated \(2\) units to the right and \(3\) units above the parent function: \(y = (\frac{1}{4})^x\). Note the differences in their graphs:

The general form of a logarithmic function is: \(y = \log_{a}x\) or \(y = ln(x)\). It graphs as the following:

Much like exponential equations, logarithmic equations can be horizontally and vertically translated by modifying the argument or adding to the function as a whole, respectively:

Interpreting

Interpreting exponential and logarithmic functions involves the ability to translate an algebraic representation of the function into its corresponding graph, and vice versa, as well as the ability to draw conclusions from the algebraic or graphical representation of the function.

As with other functions, the ability to determine the critical values and end behavior is crucial for accurately assessing a particular equation. Understanding how increasing or decreasing a variable influences the function as well as the limitations associated with a function are equally important.