Exponential and Logarithmic Equations
Solve applied problems involving exponential and logarithmic equations.
In previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen that any exponential function can be written as a logarithmic function and vice versa. We have used exponents to solve logarithmic equations and logarithms to solve exponential equations. We are now ready to combine our skills to solve equations that model real-world situations, whether the unknown is in an exponent or in the argument of a logarithm.
One such application is in science, in calculating the time it takes for half of the unstable material in a sample of a radioactive substance to decay, called its half-life . The table below lists the half-life for several of the more common radioactive substances.
We can see how widely the half-lives for these substances vary. Knowing the half-life of a substance allows us to calculate the amount remaining after a specified time. We can use the formula for radioactive decay:
- [latex]{A}_{0}[/latex] is the amount initially present
- T is the half-life of the substance
- t is the time period over which the substance is studied
- y is the amount of the substance present after time t
Example 13: Using the Formula for Radioactive Decay to Find the Quantity of a Substance
How long will it take for ten percent of a 1000-gram sample of uranium-235 to decay?
[latex]\begin{cases}\text{ }y=\text{1000}e\frac{\mathrm{ln}\left(0.5\right)}{\text{703,800,000}}t\hfill & \hfill \\ \text{ }900=1000{e}^{\frac{\mathrm{ln}\left(0.5\right)}{\text{703,800,000}}t}\hfill & \text{After 10% decays, 900 grams are left}.\hfill \\ \text{ }0.9={e}^{\frac{\mathrm{ln}\left(0.5\right)}{\text{703,800,000}}t}\hfill & \text{Divide by 1000}.\hfill \\ \mathrm{ln}\left(0.9\right)=\mathrm{ln}\left({e}^{\frac{\mathrm{ln}\left(0.5\right)}{\text{703,800,000}}t}\right)\hfill & \text{Take ln of both sides}.\hfill \\ \mathrm{ln}\left(0.9\right)=\frac{\mathrm{ln}\left(0.5\right)}{\text{703,800,000}}t\hfill & \text{ln}\left({e}^{M}\right)=M\hfill \\ \text{ }\text{ }t=\text{703,800,000}\times \frac{\mathrm{ln}\left(0.9\right)}{\mathrm{ln}\left(0.5\right)}\text{years}\begin{cases}{cccc}& & & \end{cases}\hfill & \text{Solve for }t.\hfill \\ \text{ }\text{ }t\approx \text{106,979,777 years}\hfill & \hfill \end{cases}[/latex]
Analysis of the Solution
Ten percent of 1000 grams is 100 grams. If 100 grams decay, the amount of uranium-235 remaining is 900 grams.
How long will it take before twenty percent of our 1000-gram sample of uranium-235 has decayed?
- Precalculus. Authored by : Jay Abramson, et al.. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download For Free at : http://cnx.org/contents/[email protected].
General Mathematics Quarter 1 – Module 22: Solving Real-life Problems Involving Exponential Functions, Equations, and Inequalities
This module was designed and written with you in mind. It is here to help you solve real-life problems involving exponential functions, equations, and inequalities. Most of the time, students like you ask why you need to study Mathematics. Even though you know the answer, still you keep on asking this question because perhaps you did not realize how important it is to real-life situations.
This module hopes to help you make a wise decision in the future because it involves money matter problems.
After going through this module, you are expected to:
1. recall how to solve exponential functions, equations, and inequalities; and
2. solve real-life problems involving exponential functions, equations, and inequalities.
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Exponential Functions - Problem Solving
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Complex numbers.
The beauty of Algebra through complex numbers, fractals, and Euler’s formula.
- Andrew Hayes
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
- rules of exponents - algebraic
- solving exponential equations
- graphs of exponential functions .
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 \(A+B\).
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General Mathematics Solving Real-Life Problems Involving Exponential Functions, Equations, and Inequalities An exponential function is a function of the form...
This video tutorial is about Solving real-life problems involving exponential functions, equations and inequalities with several examples
6.3: Exponential Equations and Inequalities
Quarter 1 - Module 22: Solving Real-life Problems Involving Exponential Functions, Equations, and Inequalities Introductory Message This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home.
Solve the resulting equation, S = T, for the unknown. Example 4.7.1: Solving an Exponential Equation with a Common Base. Solve 2x − 1 = 22x − 4. Solution. 2x − 1 = 22x − 4 The common base is 2 x − 1 = 2x − 4 By the one-to-one property the exponents must be equal x = 3 Solve for x. Exercise 4.7.1. Solve 52x = 53x + 2.
This topic covers: - Radicals & rational exponents - Graphs & end behavior of exponential functions - Manipulating exponential expressions using exponent properties - Exponential growth & decay - Modeling with exponential functions - Solving exponential equations - Logarithm properties - Solving logarithmic equations - Graphing logarithmic functions - Logarithmic scale
First, we have to cancel the coefficient behind the exponential function. Therefore, we divide both sides by 5: 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:
‼️FIRST QUARTER‼️🟣 GRADE 11: REPRESENTING REAL-LIFE SITUATIONS USING EXPONENTIAL FUNCTIONS‼️SHS MATHEMATICS PLAYLIST‼️🟣General MathematicsFirst Quarter: ht...
Solve applied problems involving exponential and logarithmic equations In previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen that any exponential function can be written as a logarithmic function and vice versa.
1. recall how to solve exponential functions, equations, and inequalities; and 2. solve real-life problems involving exponential functions, equations, and inequalities. genmath_q1_mod22_SolvingRealLifeProblemsInvolvingExponentialFunctionsEquationsandInequalities_v2 Share with your friends! This module was designed and written with you in mind.
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.
7 years ago. For the 2 sides of your equation to be equal, the exponents must be equal. So, you can change the equation into: -2b = -b. Then, solve for "b". Sal does something very similar at about. 3:45. in the video. Hope this helps.
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).
c 3z =9z+5 3 z = 9 z + 5 Show Solution. d 45−9x = 1 8x−2 4 5 − 9 x = 1 8 x − 2 Show Solution. Now, the equations in the previous set of examples all relied upon the fact that we were able to get the same base on both exponentials, but that just isn't always possible. Consider the following equation. 7x =9 7 x = 9.
y = 10000(1.05)t y = 10000 ( 1.05) t. We divide both sides by 10000 to isolate the exponential expression on one side. y 10000 = 1.05t y 10000 = 1.05 t. Next we rewrite this in logarithmic form to express time as a function of the accumulated future value. We'll use function notation and call this function g(y) g ( y).
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.
Study with Quizlet and memorize flashcards containing terms like most common applications in real-life of exponential functions, exponential growth, compound interest and more. ... Solving Problems involving Exponential Functions, Equations, and Exponential Inequalities. Flashcards. Learn. Test. Match. Flashcards. Learn. Test. Match. Created by.
Link Partners Clearly aligned math exercises on exponential equations and inequalities. Solve the exponential equations and exponential inequalities on Math-Exercises.com.
Using Like Bases to Solve Exponential Equations. The first technique involves two functions with like bases. Recall that the one-to-one property of exponential functions tells us that, for any real numbers b,b, S,S, and T,T, where b>0,b≠1,b>0,b≠1, bS=bTbS=bT if and only if S=T.S=T.
GENERAL MATHEMATICSTOPICS⬇️Solving Real-life Problems involving Inverse FunctionsRepresenting Real-life Situations Using Exponential FunctionsExponential Functions, Equations and Inequalities This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts. See Answer
GENERAL MATHEMATICS LEARNING ACTIVITY SHEET Solving Real-life Problems involving Inverse Functions Representing Real-life Situations Using Exponential Functions Exponential Functions, Equations and Inequalities What I Can Do Read and understand the situation below, then answer the questions or perform the tasks You have a best friend, and she is also an 18-year old senior high school student ...
Question: GENERAL MATHEMATICS LEARNING ACTIVITY SHEET Solving Real-life Problems involving Inverse Functions Representing Real-life Situations Using Exponential Functions Exponential Functions, Equations and Inequalities The predicted population for the year 2030 is 269, 971.
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Solving Exponential Equations . An exponential equation 15 is an equation that includes a variable as one of its exponents. In this section we describe two methods for solving exponential equations. First, recall that exponential functions defined by \(f (x) = b^{x}\) where \(b > 0\) and \(b ≠ 1\), are one-to-one; each value in the range corresponds to exactly one element in the domain.