## Math Skills Overview Guide

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## Word Problems- Exponential

video, practice problems.

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## Exponential Growth and Decay Word Problems

Watch a Khan Academy Video » Length: 7:22

- Interpret change in exponential models
- Exponential vs. linear models
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- Last Updated: Jul 7, 2023 12:00 PM
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Home > Math Worksheets > Algebra Worksheets > Solving Exponential Equations

These worksheets demonstrate the steps required to solve exponential equations and give practice problems to help students master the skill. We cover the common types of problems you run into with these worksheets. There are two main types of difference between exponential equations problems, and it entirely depends on the bases. If the bases are the same, the problem is most likely pretty easy. If the bases differ that can really complicate things. The first step is usually to set the exponents equal to one another when the bases are the same. Next step is to solve for the unknown variable. Once you do that your answer is within your grasp.

## Get Free Worksheets In Your Inbox!

Print solving exponential equations worksheets, click the buttons to print each worksheet and associated answer key., solving exponential equations lesson.

This starts with a full lesson how to manipulate these types of equations. Learn how to solve problems like the following: Solve the exponential equation: [25] (x+1) = 625

## Practice Worksheet 1

Solve these 10 exponential equations. Example: [15] (x+2) = 225

## Worksheet 2

you will work on solving 10 exponential equations. These problems are a bit more challenging. Example: 4 (x+3) = 256

## Solving Exponential Equations Review Sheet

Follow the steps to solve the following problem: [221] (x-5) = 49. Afterwards, practice the skill by completing six similar problems.

## Full Topic Quiz

Solve these 10 exponential equations and then score your answers. Example: [21] (x+3) = 441. A scoring key is provided at the bottom to help you track your scores.

## Do Now Pack

Complete the problems. Put your answer in the "My Answer" box. Example: 5 (5x-11) = 625

## Lacking a Common Base Lesson

Learn how to solve the following problem: Solve the exponential equation 5 d = 22

## Lacking a Common Base Worksheet

You will solve problems that do not have a set common base.

These problems go off in a more advanced type of an equation for students to work with.

## Review Section

Follow the steps to solve the following problem: Solve the exponential equation 11 d = 23. Then practice the skill by solving the problems given. This serves as a review of all the skills that we have learned.

## No Common Base Quiz

We see how students did with this skill in a informal quiz.

## No Common Base Do Now

this is a good way to either introduce or review this skill with your class. They will be able to plot how they completed the problem and how it should be done, if they fell off the path.

## How to Solve Exponential Equations

Teaching algebra to kids can be exhausting. The introduction of variables makes it difficult for them to understand equations. One of the primary challenges is to teach the students how to solve exponential equations. If your class's kids struggle with solving exponential algebraic problems, we have a few methods to help you.

Before we jump into the ways of solving exponential equations, it is vital to understand what they are. Let's look at the formal definition of exponential equations.

What Are They?

Exponential equations refer to algebraic equations with a variable in the exponent position. To solve such problems, you need to evaluate the value of exponents. For example, the equation 4 = 2x is an exponential equation.

Equating Same Bases

When you have to solve an exponential equation, see if the two sides of the equation have the same base value. In such cases, you can directly ignore the bases and evaluate exponential values.

For example, if you have to solve 43y = 46, you can ignore the same base on both sides and simplify the equation as 3y = 6.

We ignore the same base values due to their equality. If we apply logic to it, two bases with the same values on both sides of the equation are equal. We only need to work on the exponential part of the equation. By simplifying the exponents on both sides, we can evaluate the answer. You can test the accuracy of your solution by inputting the answer into the exponential variable.

Equating Exponent With Whole Number

In some cases, you may come across equations with an exponential expression on one side and a whole number on the other side of the equation. To simplify such algebraic equations, you can eliminate the exponent's base value and equate it with the whole number on the other side.

For example, if you find an equation 2y + 1 + 4 = 8, you can apply the subtraction rule on both sides to equate the exponential value against the whole number. Here is how you can simplify it:

2y + 1 + 4 - 4 = 8 - 4 2y + 1 = 4

Using this method, you can change the whole number into a similar expression as on the other side of the equation to get the value for the variable.

2y + 1 = 22

Now you can ignore the same base values on both sides and simplify the equation to find the variable's value.

Use Log for Different Bases

To apply this method, you must have all the exponential values isolated on one side of the equation and the whole number on the other. Once done, you can apply the addition or subtraction rule to both sides for simplification.

For example, if you have an equation, 4y - 2 - 8 = 8, you can apply the addition rule like this:

4y - 1 - 8 + 8 = 8 + 8 4y - 1 = 16

Since you do not have the same base values on both sides of the equation, you can apply a log on both sides.

log 4 y -1=log 16

(y-1)log 4 = log 16

Applying the division rule to both sides, you can simplify log values on both sides.

(y - 1)log 4 /log 4 =log 16 / log 4

y - 1 = log 4

Simplify it further to evaluate the value of the variable.

y - 1 + 1 = log 4 + 1

y = log 4 + 1

y = 0.6020 + 1 y = 1.6020

Using the methods mentioned above, you can solve exponential equations quickly. If you want to teach your students all three ways, you can find several exponential equations online to test their learning. You may also come up with some of your exponential algebraic equations to improve the skills of your students.

Exponents always can complicate equations, not because it complicates the concepts, but it does make the calculations more difficult for students. This series of worksheets will work on how to solve for exponential variables in algebraic expressions through the use of logarithmic tables and by balancing the equation. The complete set contains all introductory material, practice questions, reviews, longer exercise sheets, and quizzes. We will walk through the vocabulary and set you in the right direction to process these types of problems. This series of problems will normally require you to calculate and take many different thoughts into consideration before approaching problems like this. Exponents are the backbone of the problems and need to be kept in the back of your mind when you choose each step of your process. These problems often have the variable located within the exponent with heightens the difficult and conceptualization of the problem itself.

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## Unit 2: Solving equations & inequalities

About this unit, linear equations with variables on both sides.

- Why we do the same thing to both sides: Variable on both sides (Opens a modal)
- Intro to equations with variables on both sides (Opens a modal)
- Equations with variables on both sides: 20-7x=6x-6 (Opens a modal)
- Equation with variables on both sides: fractions (Opens a modal)
- Equation with the variable in the denominator (Opens a modal)
- Equations with variables on both sides Get 3 of 4 questions to level up!
- Equations with variables on both sides: decimals & fractions Get 3 of 4 questions to level up!

## Linear equations with parentheses

- Equations with parentheses (Opens a modal)
- Reasoning with linear equations (Opens a modal)
- Multi-step equations review (Opens a modal)
- Equations with parentheses Get 3 of 4 questions to level up!
- Equations with parentheses: decimals & fractions Get 3 of 4 questions to level up!
- Reasoning with linear equations Get 3 of 4 questions to level up!

## Analyzing the number of solutions to linear equations

- Number of solutions to equations (Opens a modal)
- Worked example: number of solutions to equations (Opens a modal)
- Creating an equation with no solutions (Opens a modal)
- Creating an equation with infinitely many solutions (Opens a modal)
- Number of solutions to equations Get 3 of 4 questions to level up!
- Number of solutions to equations challenge Get 3 of 4 questions to level up!

## Linear equations with unknown coefficients

- Linear equations with unknown coefficients (Opens a modal)
- Why is algebra important to learn? (Opens a modal)
- Linear equations with unknown coefficients Get 3 of 4 questions to level up!

## Multi-step inequalities

- Inequalities with variables on both sides (Opens a modal)
- Inequalities with variables on both sides (with parentheses) (Opens a modal)
- Multi-step inequalities (Opens a modal)
- Using inequalities to solve problems (Opens a modal)
- Multi-step linear inequalities Get 3 of 4 questions to level up!
- Using inequalities to solve problems Get 3 of 4 questions to level up!

## Compound inequalities

- Compound inequalities: OR (Opens a modal)
- Compound inequalities: AND (Opens a modal)
- A compound inequality with no solution (Opens a modal)
- Double inequalities (Opens a modal)
- Compound inequalities examples (Opens a modal)
- Compound inequalities review (Opens a modal)
- Solving equations & inequalities: FAQ (Opens a modal)
- Compound inequalities Get 3 of 4 questions to level up!

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## Exponential Equations And Inequalities

Exponential Equations And Inequalities - Displaying top 8 worksheets found for this concept.

Some of the worksheets for this concept are Exponential equations not requiring logarithms, Solving exponential equations, Exponential equations and inequalities, Logarithmic equations and inequalities, Solve exponential equations and inequalities answer key, Solving exponential and logarithmic equations, Infinite algebra 2, Exponential log equations.

Found worksheet you are looking for? To download/print, click on pop-out icon or print icon to worksheet to print or download. Worksheet will open in a new window. You can & download or print using the browser document reader options.

## 1. Exponential Equations Not Requiring Logarithms

2. solving exponential equations, 3. 6.3 exponential equations and inequalities, 4. 6.4 logarithmic equations and inequalities, 5. solve exponential equations and inequalities answer key, 6. solving exponential and logarithmic equations, 7. infinite algebra 2, 8. exponential & log equations.

## Solving Inequality Word Questions

(You might like to read Introduction to Inequalities and Solving Inequalities first.)

In Algebra we have "inequality" questions like:

## Sam and Alex play in the same soccer team. Last Saturday Alex scored 3 more goals than Sam, but together they scored less than 9 goals. What are the possible number of goals Alex scored?

How do we solve them?

The trick is to break the solution into two parts:

Turn the English into Algebra.

Then use Algebra to solve.

## Turning English into Algebra

To turn the English into Algebra it helps to:

- Read the whole thing first
- Do a sketch if needed
- Assign letters for the values
- Find or work out formulas

We should also write down what is actually being asked for , so we know where we are going and when we have arrived!

The best way to learn this is by example, so let's try our first example:

Assign Letters:

- the number of goals Alex scored: A
- the number of goals Sam scored: S

We know that Alex scored 3 more goals than Sam did, so: A = S + 3

And we know that together they scored less than 9 goals: S + A < 9

We are being asked for how many goals Alex might have scored: A

Sam scored less than 3 goals, which means that Sam could have scored 0, 1 or 2 goals.

Alex scored 3 more goals than Sam did, so Alex could have scored 3, 4, or 5 goals .

- When S = 0, then A = 3 and S + A = 3, and 3 < 9 is correct
- When S = 1, then A = 4 and S + A = 5, and 5 < 9 is correct
- When S = 2, then A = 5 and S + A = 7, and 7 < 9 is correct
- (But when S = 3, then A = 6 and S + A = 9, and 9 < 9 is incorrect)

## Lots More Examples!

## Example: Of 8 pups, there are more girls than boys. How many girl pups could there be?

- the number of girls: g
- the number of boys: b

We know that there are 8 pups, so: g + b = 8, which can be rearranged to

We also know there are more girls than boys, so:

We are being asked for the number of girl pups: g

So there could be 5, 6, 7 or 8 girl pups.

Could there be 8 girl pups? Then there would be no boys at all, and the question isn't clear on that point (sometimes questions are like that).

- When g = 8, then b = 0 and g > b is correct (but is b = 0 allowed?)
- When g = 7, then b = 1 and g > b is correct
- When g = 6, then b = 2 and g > b is correct
- When g = 5, then b = 3 and g > b is correct
- (But if g = 4, then b = 4 and g > b is incorrect)

A speedy example:

## Example: Joe enters a race where he has to cycle and run. He cycles a distance of 25 km, and then runs for 20 km. His average running speed is half of his average cycling speed. Joe completes the race in less than 2½ hours, what can we say about his average speeds?

- Average running speed: s
- So average cycling speed: 2s
- Speed = Distance Time
- Which can be rearranged to: Time = Distance Speed

We are being asked for his average speeds: s and 2s

The race is divided into two parts:

- Distance = 25 km
- Average speed = 2s km/h
- So Time = Distance Average Speed = 25 2s hours
- Distance = 20 km
- Average speed = s km/h
- So Time = Distance Average Speed = 20 s hours

Joe completes the race in less than 2½ hours

- The total time < 2½
- 25 2s + 20 s < 2½

So his average speed running is greater than 13 km/h and his average speed cycling is greater than 26 km/h

In this example we get to use two inequalities at once:

## Example: The velocity v m/s of a ball thrown directly up in the air is given by v = 20 − 10t , where t is the time in seconds. At what times will the velocity be between 10 m/s and 15 m/s?

- velocity in m/s: v
- the time in seconds: t
- v = 20 − 10t

We are being asked for the time t when v is between 5 and 15 m/s:

So the velocity is between 10 m/s and 15 m/s between 0.5 and 1 second after.

And a reasonably hard example to finish with:

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- How do you solve exponential inequalities?
- To solve exponential inequalities, write the inequality in the form of a single exponential function with a positive base. Take the logarithm of both sides of the inequality. Simplify the logarithmic expression to isolate the variable. Solve the inequality for the variable. Check the solution by plugging it back into the original inequality.
- What is exponential inequality?
- An exponential inequality is an inequality that involves one or more exponents.
- How do you solve exponential inequalities with fractions?
- To solve exponential inequalities with fractions, first convert the fractions to powers of the base using the properties of exponents. Then, apply the usual methods for solving exponential inequalities.

exponential-inequalities-calculator

- 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... Read More

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Course: Algebra 2 > Math > Algebra 2 > Modeling > Equations & inequalities word problems Google Classroom The Smiths and the Johnsons were competing in the final leg of the Amazing Race. In their race to the finish, the Smiths immediately took off on a 165 kilometer path traveling at an average speed of v kilometers per hour.

Solve the exponential inequalities : You might be also interested in: - Exponential Function - Linear Equations and Inequalities - Systems of Equations and Inequalities - Quadratic Equations and Inequalities - Irrational Equations and Inequalities - Logarithmic Equations and Inequalities - Trigonometric Equations and Inequalities

7-2 Solving Exponential Equations and Inequalities Word Problem Practice.pdf.

Here is a set of practice problems to accompany the Solving Exponential Equations section of the Exponential and Logarithm Functions chapter of the notes for Paul Dawkins Algebra course at Lamar University.

We summarize below the two common ways to solve exponential equations, motivated by our examples. Steps for Solving an Equation involving Exponential Functions. Isolate the exponential function. If convenient, express both sides with a common base and equate the exponents.

Exercise 4.6e. 5. ★ For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation. 121. log( 1 100) = − 2. 122. log324(18) = 1 2. ★ For the following exercises, use the definition of a logarithm to solve the equation. 123. 5log7n = 10.

Equations & Inequalities; Logarithms Toggle Dropdown. Introduction to Logarithms ; ... Exponential Growth and Decay Word Problems. Watch a Khan Academy Video » Length: 7:22 . Practice Problems Interpret change in exponential models. Exponential vs. linear models << Previous: Writing ...

Quadratic and exponential word problems ask us to solve equations or evaluate functions that model real-world scenarios using quadratic and exponential expressions. On your official SAT, you'll likely see 1 to 2 questions that are quadratic or exponential word problems. This lesson builds on an understanding of the following skills: Solving ...

Practice Exponential Equations, receive helpful hints, take a quiz, improve your math skills. ... Equations Rational Equations Radical Equations Logarithmic Equations Exponential Equations Absolute Equations Polynomials Inequalities System of Equations. Matrices & Vectors. ... Word Problems. Word Problems.

Solve Exponential & Logarithmic Equations & Word Problems. Created by. MathHop by Jackie B. Exponential & Logarithmic Equations & Word Problems: This worksheet provides practice solving exponential & logarithmic equations, including word problems that require the use of logarithms. There are 18 exponential equations to solve.

Inequalities word problems. Google Classroom. Kwame must earn more than 16 stars per day to get a prize from the classroom treasure box. Write an inequality that describes S , the number of stars Kwame must earn per day to get a prize from the classroom treasure box.

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Compound Interest. If we put P P dollars into an account that earns interest at a rate of r r (written as a decimal as opposed to the standard percent) for t t years then, if interest is compounded m m times per year we will have, A =P (1 + r m)tm A = P ( 1 + r m) t m dollars after t t years.

Exponential equations refer to algebraic equations with a variable in the exponent position. To solve such problems, you need to evaluate the value of exponents. For example, the equation 4 = 2x is an exponential equation. Equating Same Bases. When you have to solve an exponential equation, see if the two sides of the equation have the same ...

Let's explore some different ways to solve equations and inequalities. We'll also see what it takes for an equation to have no solution, or infinite solutions. Linear equations with variables on both sides Learn Why we do the same thing to both sides: Variable on both sides Intro to equations with variables on both sides

Here are a set of practice problems for the Solving Equations and Inequalities chapter of the Algebra notes. If you'd like a pdf document containing the solutions the download tab above contains links to pdf's containing the solutions for the full book, chapter and section. At this time, I do not offer pdf's for solutions to individual problems.

Exponential Equations And Inequalities - Displaying top 8 worksheets found for this concept.. Some of the worksheets for this concept are Exponential equations not requiring logarithms, Solving exponential equations, Exponential equations and inequalities, Logarithmic equations and inequalities, Solve exponential equations and inequalities answer key, Solving exponential and logarithmic ...

Symbolab is the best calculator for solving a wide range of word problems, including age problems, distance problems, cost problems, investments problems, number problems, and percent problems. Show more Why users love our Word Problems Calculator Middle School Math Solutions - Inequalities Calculator

Exponential Inequalities: Problems with Solutions By Denitsa Dimitrova (Bulgaria) Problem 1 \displaystyle 5^ {x^2+3} \le 5^ {4x} 5x2+3 ≤ 54x \displaystyle [1, 3] [1,3] \displaystyle (-\infin, 1]\cup [3,+\infin) (−∞,1]∪[3,+∞) \displaystyle [2, 9] [2,9] \displaystyle (-\infin, 2]\cup [9,+\infin) (−∞,2]∪[9,+∞) Problem 2

Simplify: (W − 4)2 ≤ 9. Take the square root on both sides of the inequality: −3 ≤ W − 4 ≤ 3. Yes we have two inequalities, because 32 = 9 AND (−3)2 = 9. Add 4 to both sides of each inequality: 1 ≤ W ≤ 7. So the width must be between 1 m and 7 m (inclusive) and the length is 8−width.

To solve exponential inequalities, write the inequality in the form of a single exponential function with a positive base. Take the logarithm of both sides of the inequality. Simplify the logarithmic expression to isolate the variable. Solve the inequality for the variable. Check the solution by plugging it back into the original inequality.

In the unknown two-digit number is its units digit less by one than its tens digit. If we add 7 to this number, the new number will be greater than 19, while less than 51. Find all two-digit numbers with those characteristics. Find the fraction about which we can say: If the denominator is reduced by one, fraction equals to ½.

Our objective in solving 75 = 100 1 + 3e − 2t is to first isolate the exponential. To that end, we clear denominators and get 75(1 + 3e − 2t) = 100. From this we get 75 + 225e − 2t = 100, which leads to 225e − 2t = 25, and finally, e − 2t = 1 9. Taking the natural log of both sides gives ln(e − 2t) = ln(1 9).