Exponential Inequalities

An exponential inequality is one in which a variable occurs in the exponent.

As was done with solving exponential equations, we will examine
two algebraic methods for solving exponential inequalities:
Method 1: Using a Common Base  and  Method 2: Using Logarithms

A graphical solution will be offered as a check on the algebraic solutions.

Things to keep in mind when working with exponential inequalities:

• an exponential function bxis always positive.    (bx < 0 has no solution) • an exponential function is increasing when b > 1. • an exponential function is decreasing when 0 < b < 1. • as was done in linear inequalities, multiplying (or dividing) by a negative value reverses the inequality sign.

Method 1: Common (same) Base If the bases are the same (or can be rewritten to be the same), set up an inequality using the exponents.

For exponential inequalities with the same base: (1) For b > 1,    bx > by   if and only if  x > y. (2) For 0 < b < 1,   bx > by   if and only if  x < y.

* Inequality Direction: If b > 1, direction unchanged If 0 < b < 1, direction flips

Let's take a look a couple of examples, and the graphs to illustrate the answers.

4x-3 > 4-1 • The base 4 is greater than 0, so "keep the direction" of the inequality. • Set up the exponents with a "greater than" symbol:     x - 3 > -1 • Solve for x:   x > 2        In interval notation: (2,∞)
(0.5)2x > (0.5)5 • The base 0.5 is in the interval 0 < x < 1, so "reverse the direction" of the inequality. • Set up the exponents with a "less than" symbol (flip):   2x < 5 • Solve for x:  x < 5/2     In interval notation: (-∞, 5/2)

Method 2: Bases Not the Same If the bases are not the same, take the logarithm of both sides.

For exponential inequalities with different bases, take the logarithm of both sides. Choose the logarithm "base" such that an inverse relationship can be applied to simplify the inequality.
There is no flipping of the inequality in these solutions.

3x+2 > 10 • The bases are not the same, (and cannot easily be "made" the same), so use logs. • Choose log3 as it will allow an inverse relationship with the exponential 3x+2 . • Apply log3 to both sides:       log3 3x+2 > log3 10. • Apply inverse relationship on the left side:       x + 2 > log3 10. • Solve for x:   x > 2.09590327429 - 2    and x > 0.096.     In interval notation: (0.096,∞)
ex < 4.5 • The bases are not the same, so use logs. • Choose ln as it will allow an inverse relationship with the exponential ex. • Apply ln to both sides:     ln( ex ) < ln(4.5). • Apply inverse relationship on the left side:       x < ln(4.5). • Solve for x:   x < 1.504077397    and    x < 1.504             In interval notation: (-∞,1.504]

Logarithmic Inequalities

A logarithmic inequality is simply an inequality containing one (or more) logarithms with variable expressions.

Things to keep in mind when working with logarithmic inequalities:

• a logarithm's domain is positive • use exponentiation determined by the log's base. • if the base is greater than 1, the inequality direction (sign) remains the same. • if the base is between 0 and 1, the inequality direction (sign) flips.