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Exponential Decay
Negative Exponents
Multiplying and Dividing Fractions 4
Evaluating Expressions Involving Fractions
The Cartesian Coordinate System
Adding and Subtracting Fractions with Like Denominators
Solving Absolute Value Inequalities
Multiplying Special Polynomials
FOIL Method
Inequalities
Solving Systems of Equations by Graphing
Graphing Compound Inequalities
Solving Quadratic Equations by Completing the Square
Addition Property of Equality
Square Roots
Adding and Subtracting Fractions
The Distance Formula
Graphing Logarithmic Functions
Fractions
Dividing Mixed Numbers
Evaluating Polynomials
Power of a Product Property of Exponents
Terminology of Algebraic Expressions
Adding and Subtracting Rational Expressions with Identical Denominators
Solving Exponential Equations
Factoring The Difference of 2 Squares
Changing Fractions to Decimals
Solving Linear Equations
Using Patterns to Multiply Two Binomials
Completing the Square
Roots of Complex Numbers
Methods for Solving Quadratic Equations
Conics in Standard Form
Solving Quadratic Equations by Using the Quadratic Formula
Simplifying Fractions 2
Exponential Notation
Exponential Growth
The Cartesian Plane
Graphing Linear Functions
The Slope of a Line
Finding Cube Roots of Large Numbers
Rotating Axes
Common Mistakes With Percents
Solving an Equation That Contains a Square Root
Rational Equations
Properties of Common Logs
Composition of Functions
Using Percent Equations
Solving Inequalities
Properties of Exponents
Graphing Quadratic Functions
Factoring a Polynomial by Finding the GCF
The Rectangular Coordinate System
Adding and Subtracting Fractions
Multiplying and Dividing Rational Expressions
Improper Fractions and Mixed Numbers
Properties of Exponents
Complex Solutions of Quadratic Equations
Solving Nonlinear Equations by Factoring
Solving Quadratic Equations by Factoring
Least Common Multiples
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Solving Exponential Equations
Solving Linear Equations
Multiplication Property of Equality
Multiplying Mixed Numbers
Multiplying Fractions
Reducing a Rational Expression to Lowest Terms
Literal Numbers
Factoring Trinomials
Logarithmic Functions
Adding Fractions with Unlike Denominators
Simplifying Square Roots
Adding Fractions
Equations Quadratic in Form
Dividing Rational Expressions
Slopes of Parallel Lines
Simplifying Cube Roots That Contain Variables
Functions and Graphs
Complex Numbers
Multiplying and Dividing Fractions 1
Composition of Functions
Intercepts of a Line
Powers
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Multiplying Two Numbers with the same Tens Digit and whose Ones Digits add up to 10
Factoring Trinomials
Exponents and Polynomials
Decimals and their Equivalent Fractions
Negative Integer Exponents
Adding and Subtracting Mixed Numbers
Solving Quadratic Equations
Theorem of Pythagoras
Equations 1
Subtracting Fractions
Solving Quadratic Equations by Graphing
Evaluating Polynomials
Slope
Angles and Degree Measure
   
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Solving Exponential Equations

An exponential equation is an equation where one or more of the exponents contains a variable. Some types of exponential equations can be solved using the following property.

 

Property — Exponential Equality

If bx = by, then x = y. Here, b > 0 and b 1.

Note:

Recall that for a one-to-one function, two different inputs always result in two different outputs.

That is, in a one-to-one function, each output value corresponds to exactly one input value.

This property is a direct consequence of the fact that exponential functions are one-to-one functions. That is, if two exponential functions have the same output, bx and by, then their inputs, x and y, must be equal.

 

Example 1

Solve: 25 5x-8 = 625

Solution

First, write each expression using the same base, 5.

On the left, use the Power of a Power Property.

Use the Exponential Equality Property to set the exponents equal to each other.

25 5x - 8

(52) 5x - 8

5 10x - 16

10x - 16

= 625

= 54

= 54

= 4

Add 16 to both sides.

Divide both sides by 10.

10x

x

= 20

= 2

We can check the solution by replacing x with 2 in the original equation and simplifying.

Original equation:

 

Is

Is

Is

25 5x - 8

25 5 · 2 - 8

252

625

= 625

= 625 ?

= 625 ?

= 625 ? Yes

So, the solution of 25 5x - 8 = 625 is x = 2.

Note:

To write each base in exponential form, first find its prime factorization.

For example:

25 = 5 · 5 = 52

625 = 5 · 5 · 5 · 5 = 54

The Power of a Power Property says (xm)n 5 xmn.

 

Example 2

Solve: 163x + 1 = 324x

Solution

Write each expression with the same base, 2.

Use the Power of a Power Property.

Use the Exponential Equality Property to set the exponents equal to each other.

163x + 1

(24)3x + 1

212x + 4

12x + 4

= 324x

= (25)4x

= 220x

= 20x

Subtract 12x from both sides. 4 = 8x
Divide both sides by 8. = x

We can check the solution in the usual way.

So, the solution of 163x + 1 = 324x is

Note:

This method can only be used if we write each side of the equation as an exponential expression using the same base. If not, we must use a different method.

For example, we can use this method solve 2x = 8 but not to solve 2x = 9.

Recall that

Thus,

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