High School Math Solutions – Inequalities Calculator, Exponential Inequalities

Last post, we talked about how to solve logarithmic inequalities. This post, we will learn how to solve exponential inequalities. The method of solving exponential inequalities is very similar to solving logarithmic inequalities. In these problems, the variable is in the exponent and our goal is to isolate the variable, which means getting it out of the exponent. We will see some these problems in the forms:

                                                                   e^x>a

                                                                   a^x<b

Let’s see how to solve these problems step by step.

1. Use algebraic manipulation to move everything that is not in the exponential expression to one side
2. Isolate the variable by getting rid of the exponential expression

Example:

e^x>a
e^x>e^{\ln(⁡a)}
x>\ln(a)

Example:

a^x<b
a^x<a^{\log_a(b)}
x<\log_a⁡(b)

3. Solve the inequality

These problems are a little simpler than solving logarithmic inequalities. Let’s see how to solve an example step by step.

First example (click here):

                                                     3^{x+1}+1<2

Step 1: Use algebraic manipulation to move everything that is not in the exponential expression to one side

                                                    3^{2x+1}+1<2

                                                       3^{2x+1}<1

Step 2: Isolate the variable by getting rid of the exponential expression

                                                       3^{2x+1}<1

                                                  3^{2x+1}<3^{\log_3(1)}

                                                       2x+1<\log_3⁡(1)

Step 3: Solve the inequality

                                                       2x+1<\log_3(1)

                                                           2x+1<0

                                                            x<\frac{-1}{2}

That was pretty simple. Let’s see some more examples.

Second example (click here):

Last example (click here):

Solving exponential inequalities are simpler than solving logarithmic inequalities. However, it can still get a little tricky when solving these inequalities with more parts. For more help and practice on solving exponential inequalities, visit Symbolab’s practice.

Until next time,

Leah