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## Course: Algebra 2 > Unit 12

## Exponential equation word problem

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## Video transcript

## How Do You Solve a Word Problem with Exponential Growth?

## Background Tutorials

Introduction to algebraic expressions.

## How Do You Evaluate an Algebraic Expression?

## Order of Operations

## What's the Order of Operations?

## Percents and Percent Proportions

## What's a Percent?

## Identifying and Evaluating Exponential Functions

## What's an Exponential Function?

## Exponential Growth

## What is Exponential Growth?

## Further Exploration

## How Do You Solve a Word Problem with Exponential Decay?

- Common Math Verbs
- Word Problems
- Math Anxiety
- Learning Math
- Preparing for Tests
- The Guide Sections
- Order of Operations
- Calculating Percentages
- Dividing Fractions
- Transposition
- Inequalities
- Laws of Exponents
- Absolute Value
- Significant Figures
- Rational Expressions and Non-Permissible Values
- Simplifying Rational Expressions
- Multiplying Rational Expressions
- Dividing Rational Expressions
- Adding and Subtracting Rational Expressions
- Rational Equations and Problem Solving
- Rational Expressions Quiz
- Defining Logarithms
- Solving Exponential Equations
- Solving Problems Involving Exponential Functions
- Solving Problems Involving Logarithmic Functions
- Factoring Quadratic Equations
- Solving Quadratic Equations by Completing the Square
- The Quadratic Formula
- Arithmetic Sequences
- Arithmetic Series
- Geometric Sequences
- Geometric Series
- Understanding Sets and Set Notations
- Solving problems Involving Sets
- Odds and Probability
- Mutually Exclusive and Non-Mutually Exclusive Events
- Dependent and Independent Events
- The fundamental Counting Principle
- Permutations
- Combinations
- Angles in Standard Position
- Trigonometric Ratios
- Transformations of Functions
- Composition of Functions
- Operations with Functions
- Inverse Functions
- Limit of a Function
- Properties of Limits
- Additional Evaluation Techniques
- Function of a Derivative
- Derivative Rules
- Trigonometric Functions
- Exponential and Logarithmic Functions
- Equation of Tangent Line
- Implicit Derivatives
- The Antiderivative
- Normal Distribution
- Central Limit Theorem
- Confidence Intervals
- Sample Size
- Hypothesis Testing
- Hypothesis Testing Process
- Additional Resources

## Attribution

## Solving Problems Involving Exponential Equations

## We need to isolate x, to do so, we have to cancel the power of x :

## In some other cases, one has to write an exponential function in a different base.

Example: convert to base , we substitute into the original equation:.

## we multiply the exponents on the right-hand-side of the equation:

## The base of the exponential function on the left-hand-side is 3 , therefore, we try to write the right-hand-side in base 3 :

## since the bases on the right-hand-side are the same, we add their exponents:

## the bases are equal on both sides of the equation, so, the exponents are also equal:

- << Previous: Solving Exponential Equations
- Next: Solving Problems Involving Logarithmic Functions >>
- Last Updated: Feb 23, 2023 11:25 AM
- URL: https://libguides.nwpolytech.ca/math

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## Solving Word Problems Involving Applications of Exponential Functions to Growth and Decay

The population of a certain county can be modeled by the equation:

Population doubles every 34.66 yrs

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## Exponential Functions - Problem Solving

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## Recommended Course

The beauty of Algebra through complex numbers, fractals, and Euler’s formula.

To solve problems on this page, you should be familiar with

## Growth and Decay

Problem solving - basic, problem solving - intermediate, problem solving - advanced.

Suppose that the population of rabbits increases by 1.5 times a month. When the initial population is 100, what is the approximate integer population after a year? The population after \(n\) months is given by \(100 \times 1.5^n.\) Therefore, the approximate population after a year is \[100 \times 1.5^{12} \approx 100 \times 129.75 = 12975. \ _\square \]

Suppose that the population of rabbits increases by 1.5 times a month. At the end of a month, 10 rabbits immigrate in. When the initial population is 100, what is the approximate integer population after a year? Let \(p(n)\) be the population after \(n\) months. Then \[p(n+2) = 1.5 p(n+1) + 10\] and \[p(n+1) = 1.5 p(n) + 10,\] from which we have \[p(n+2) - p(n+1) = 1.5 \big(p(n+1) - p(n)\big).\] Then the population after \(n\) months is given by \[p(0) + \big(p(1) - p(0)\big) \frac{1.5^{n} - 1}{1.5 - 1} .\] Therefore, the population after a year is given by \[\begin{align} 100 + (160 - 100) \frac{1.5^{12} - 1}{1.5 - 1} \approx& 100 + 60 \times 257.493 \\ \approx& 15550. \ _\square \end{align}\]

Suppose that the annual interest is 3 %. When the initial balance is 1,000 dollars, how many years would it take to have 10,000 dollars? The balance after \(n\) years is given by \(1000 \times 1.03^n.\) To have the balance 10,000 dollars, we need \[\begin{align} 1000 \times 1.03^n \ge& 10000 \\ 1.03^n \ge& 10\\ n \log_{10}{1.03} \ge& 1 \\ n \ge& 77.898\dots. \end{align}\] Therefore, it would take 78 years. \( _\square \)

The half-life of carbon-14 is approximately 5730 years. Humans began agriculture approximately ten thousand years ago. If we had 1 kg of carbon-14 at that moment, how much carbon-14 in grams would we have now? The weight of carbon-14 after \(n\) years is given by \(1000 \times \left( \frac{1}{2} \right)^{\frac{n}{5730}}\) in grams. Therefore, the weight after 10000 years is given by \[1000 \times \left( \frac{1}{2} \right)^{\frac{10000}{5730}} \approx 1000 \times 0.298 = 298.\] Therefore, we would have approximately 298 g. \( _\square \)

Given three numbers such that \( 0 < a < b < c < 1\), define

\[ A = a^{a}b^{b}c^{c}, \quad B = a^{a}b^{c}c^{b} , \quad C = a^{b}b^{c}c^{a}. \]

How do the values of \(A, B, C \) compare to each other?

\[\large 2^{x} = 3^{y} = 12^{z} \]

If \(5^x = 6^y = 30^7\), then what is the value of \( \frac{ xy}{x+y} \)?

If \(27^{x} = 64^{y} = 125^{z} = 60\), find the value of \(\large\frac{2013xyz}{xy+yz+xz}\).

\[\large f(x)=\frac{e^x+e^{-x}}{e^x-e^{-x}} \]

\[\large \left(1+\frac{1}{x}\right)^{x+1}=\left(1+\frac{1}{2000}\right)^{2000}\]

Given that \(x\) is an integer that satisfies the equation above, find the value of \(x\).

\[\Large a^{(a-1)^{(a-2)}} = a^{a^2-3a+2}\]

Find the sum of all positive integers \(a\) that satisfy the equation above.

Find the sum of all solutions to the equation

\[ \large (x^2+5x+5)^{x^2-10x+21}=1 .\]

\[\large |x|^{(x^2-x-2)} < 1 \]

If the solution to the inequality above is \(x\in (A,B) \), then find the value of \(A+B\).

## Master concepts like these

Learn more in our Complex Numbers course, built by experts for you.

## IMAGES

## VIDEO

## COMMENTS

It grows 20% every year. So this is how much he started the year with, and then he gets another 20% of that 6,250. If we factor out a 6,250, this is equal to

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