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Course: Algebra 2   >   Unit 12

Exponential equation word problem

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Video transcript

word problems involving exponential equations

How Do You Solve a Word Problem with Exponential Growth?

If something increases at a constant rate, you may have exponential growth on your hands. In this tutorial, learn how to turn a word problem into an exponential growth function. Then, solve the function and get the answer!

Background Tutorials

Introduction to algebraic expressions.

How Do You Evaluate an Algebraic Expression?

How Do You Evaluate an Algebraic Expression?

Plugging variables into an expression is essential for solving many algebra problems. See how to plug in variable values by watching this tutorial.

Order of Operations

What's the Order of Operations?

What's the Order of Operations?

Check out this tutorial where you'll see exactly what order you need to follow when you simplify expressions. You'll also see what happens when you don't follow these rules, and you'll find out why order of operations is so important!

Percents and Percent Proportions

What's a Percent?

What's a Percent?

Sales tax, tips at restaurants, grades on tests... no matter what you do, you can't run away from percents. So watch this tutorial and see once and for all what percents are all about!

Identifying and Evaluating Exponential Functions

What's an Exponential Function?

What's an Exponential Function?

Looking at an equation with a variable in the exponent? You have an exponential function! Learn about exponential functions in this tutorial.

Exponential Growth

What is Exponential Growth?

What is Exponential Growth?

Exponential functions often involve the rate of increase or decrease of something. When it's a rate of increase, you have an exponential growth function! Check out these kinds of exponential functions in this tutorial!

Further Exploration

Exponential decay.

How Do You Solve a Word Problem with Exponential Decay?

How Do You Solve a Word Problem with Exponential Decay?

If something decreases in value at a constant rate, you may have exponential decay on your hands. In this tutorial, learn how to turn a word problem into an exponential decay function. Then, solve the function and get the answer!

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Attribution

Some of the content of this guide was modeled after a guide  originally created  by the Openstax  and has been adapted for the GPRC Learning Commons in October 2020. The graphs are generated using Desmos . This work is licensed under a Creative Commons BY 4.0 International License . 

word problems involving exponential equations

Solving Problems Involving Exponential Equations

In some cases, we have to solve equations that include an exponential function where the base of the function is the variable. , example:  solve , first, we have to cancel the coefficient behind the exponential function. therefore, we divide both sides by 5:.

word problems involving exponential equations

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

word problems involving exponential equations

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:.

word problems involving exponential equations

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

word problems involving exponential equations

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 :

word problems involving exponential equations

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

word problems involving exponential equations

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

word problems involving exponential equations

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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:

P(t)=50e^{0.02t}

100 million

\frac{100,000,000}{50}=\frac{50e^{0.02t}}{50}

200 million

200,000,000=50e^{0.02t}

400 million

ln(8,000,000)=ln(e^{0.02t})

Population doubles every 34.66 yrs

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Exponential Functions - Problem Solving

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Recommended Course

Complex numbers.

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

An exponential function is a function of the form \(f(x)=a \cdot b^x,\) where \(a\) and \(b\) are real numbers and \(b\) is positive. Exponential functions are used to model relationships with exponential growth or decay. Exponential growth occurs when a function's rate of change is proportional to the function's current value. Whenever an exponential function is decreasing, this is often referred to as exponential decay .

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 the equation above is fulfilled for non-zero values of \(x,y,z,\) find the value of \(\frac { z(x+2y) }{ xy }\).

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 \(\large\frac{2013xyz}{xy+yz+xz}\).

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

Suppose we define the function \(f(x) \) as above. If \(f(a)=\frac{5}{3}\) and \(f(b)=\frac{7}{5},\) what is the value of \(f(a+b)?\)

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

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 \(A+B\).

word problems involving exponential equations

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COMMENTS

  1. Exponential equation word problem (video)

    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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  6. How Do You Solve a Word Problem with Exponential Growth?

    Exponential functions often involve the rate of increase or decrease of something. When it's a rate of increase, you have an exponential growth function! Check

  7. Solving Problems Involving Exponential Functions

    Solving Problems Involving Exponential Equations · In some cases, we have to solve equations that include an exponential function where the base

  8. Solving Word Problems Involving Applications of Exponential

    Solving Word Problems Involving Applications of Exponential Functions to Growth and Decay ... is the number of years since 1900. Find when the population is 100

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    Exponential Growth and Decay Word Problems. Write an equation for each situation and answer the question. key. (1) Bacteria can multiply at an alarming rate

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    Exponential functions are used to model relationships with exponential growth or decay. Exponential growth occurs when a function's rate of change is