Exponential and Logarithmic Inequalities

Exponential inequalities are inequalities with a variable exponent on one (or both) sides. They’re handy in circumstances that require repeated multiplication, especially when compared to a constant amount, such as interest. For example, exponential inequalities can be used to calculate how long it will take to double one’s money at a certain rate of interest; for example, with a constant interest rate of 6%, money will double in about 12 years.

Definition

Let’s start with basic definitions to distinguish between exponential and logarithmic inequality. Exponents are used in exponential inequalities, while logarithms are used in logarithmic inequalities. Both employ one of four inequalities. More than, greater than or equal to, less than, and less than or equal to are the words used.

Let’s solve a comparable math problem by starting with the answer before providing a solution method. Sounds intriguing, right? What is 2⁵? Answer: 2(2)(2)(2)(2) Equals 32. The answer is yes, but what is the question?

The problem is to find x such that 2^x > 32.

Because 2⁵ equals 32, any x bigger than 5 will suffice.

Now for the way of solution:

Step 1: Substitute an equal sign for the inequality.

Step 2: Use logarithms with exponents.

Step 3: Solve.

Step 4: Evaluate.

Step 5: Choose a domain name.

Step 6: Plot (this is an optional step).

Logarithmic inequalities

Inverse operations include logarithms and exponentials. To put it another way, one process reverses the other. For example, log 100 = 2 and 10² = 100. In addition, 10^logx Equals x. The log is in base 10.

Calculate the x values for log x².

Step 1: Replace the inequality with an equal sign.

Step 2: Raise to the power of the base using a logarithm.

Step 3: Solve the problem.

Step 4: Assess.

Step 5: Pick a name for your website.

Step 6: Plot.

System of inequality

A set of two or more inequalities in one or more variables is known as a system of inequalities. When a problem demands a variety of solutions and there are multiple constraints on those answers, systems of inequalities are used.

Systems of inequalities, like the example above, are frequently used to specify the constraints on a solution. When a problem necessitates the selection of an optimal solution, linear programming or other optimization approaches are required (combinatorial optimization or optimization with calculus).

Graphing Systems of Linear Inequalities

To graph, a two-variable linear inequality (say, x and y), start with y on one side. Consider the equation that results when the inequality sign is replaced with an equality sign. This equation’s graph is a line.

Draw a dashed line if the inequality is strict (or >). Graph a solid line if the inequality is not strict (or).

Finally, choose a point that is not on either line ((0,0) is usually the easiest) and determine whether or not the coordinates fulfil the inequality. Shade the half-plane containing that point if they do. Shade the other half-plane if they don’t.

Graph each of the system’s inequalities in the same way. The intersection region of all the solutions in the system is the solution of the system of inequalities.

Systems of linear inequalities

Let’s define inequality before we solve systems of linear inequalities. The term inequality refers to a mathematical equation with sides that are not equal.

In general, five inequality symbols are used to denote inequality equations.

Less than (<), greater than (>), less than or equal (≤), greater than or equal (≥), and not equal (≠) are the symbols used. Inequalities are used to compare numbers and find the range or ranges of values that satisfy a variable’s criteria.

What is systems of linear inequalities

A set of linear inequalities equations with the same variables is known as a system of linear inequalities.

The system of linear inequalities is solved using several methods for solving systems of linear equations. However, calculating a system of linear inequalities differs from solving linear equations because the inequality signs make the substitution and elimination methods impossible. Graphing inequalities may be the best way for solving systems of linear inequalities.

Exponential Inequalities examples

Example 1: Solve 27 / (3^-x) = 3⁶.

Solution:  know that 27  equals 3³. We may use this to make the bases identical on both sides.

3³/ (3^-x) = 3⁶

Am/an = am – n, using the quotient property of exponents. Using this method, the bases on both sides are now identical. As a result, we can make the exponents equal.

3^3-(-x) = 3⁶

3^3+x = 3⁶

3 + x = 6

By subtracting three from either side,

X = 3

As a result, the provided exponential equation’s solution is x = 3.

Example 2: Solve the 7^3x+7 = 490 exponential problem.

Solution: The number 490 cannot be written as a power of seven. As a result, the bases cannot be the same here. So we use logarithms to solve this exponential problem.

Log 7^3x+7 = log 490

Using a property of logarithms, log am = m log an is applied to both sides of the given equation. With this,

(3x + 7) log 7 = log 490 … (1)

Here, 490 = 49 × 10 = 7² × 10.

So, log 490 = log (7² × 10)

= log 7² + log 10 (because log (mn) = log m + log n)

= 2 log 7 + 1 (because log a^m = m log a and log 10 = 1)

In place of (1),

(3x + 7) log 7 = 2 log 7 + 1

Log 7 is used to divide both sides.

3x + 7 = (2 log 7 + 1) / (log 7)

3x + 7 = 2 + (1 / log 7)

7 off both sides

3x = -5 + (1 / log 7)

Dividing both sides by 3,

X = -5/3 + (1 / (3 log 7)).

Example 3: $20,000 is compounded annually at 8% annual interest, how long does it take for it to double? Your answer should be rounded to the nearest integer.

Solution: P = $20000 is the main amount.

The interest rate is r = 8%, which equals 8/100 = 0.08.

A = 20,000 x 2 = $40,000 is the final figure.

Assume that the required amount of time is t years.

When compounded annually, using the compound interest calculation,

A = P (1 + r) t

40000 = 20000 (1 + 0.08)t

Dividing both sides by 20000,

2 = (1.08)t

Taking log on both sides,

Log 2 = log (1.08)t

Log 2 = t log (1.08)

T = (log 2) / (log 1.08)

T = 9

The final answer is rounded to the nearest integer.

Example 4: Solve the following equation:

10^x = 175

Possible Answers:

X ≈ 22.4

X ≈ 2.24

X ≈ 4.48

X ≈ 0.224

X = 2.24

Solution: To solve this equation, recall the following property:

Log_by = x     Can be rewritten as b^x = y.

10^x = 175

Log₁₀175 = x

Evaluate with your calculator to get

X ≈ 2.24

Example 5: Solve

10^3x = 100

Possible Answers:

X = 10

X = 2

X = 3

X = 2/3

X = 3/2

Solution:  log (which is just log₁₀, by convention) to solve.

Log(10^3x) = log(100)

Log(10^3x) = log(10²)

3x = 2

X = 2/3.

Conclusion

Exponential inequalities are inequalities with a variable exponent on one (or both) sides. They’re handy in circumstances that require repeated multiplication, especially when compared to a constant amount, such as interest.