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6.6 Exponential and Logarithmic Equations

2 min read•june 24, 2024

Exponential equations are a powerful tool in algebra, allowing us to model growth, decay, and other real-world phenomena. These equations involve variables in exponents, and solving them often requires special techniques or the use of logarithms.

Logarithms are the inverse of exponential functions, making them crucial for solving complex exponential equations. By understanding the and their relationship to exponents, we can tackle a wide range of problems in science, finance, and other fields.

Exponential Equations

Solving exponential equations with like bases

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  • Set exponents equal when bases are the same (2x=2y2^x = 2^y means x=yx = y)
  • Solve resulting equation for the variable
    • 5x=52x−15^x = 5^{2x - 1} becomes x=2x−1x = 2x - 1, simplify to 1=x1 = x

Logarithms for exponential equations

  • Logarithms undo exponential functions (log⁡2(23)=3\log_2(2^3) = 3)
  • Isolate exponential expression on one side
  • Apply with same base to both sides
    • 4x=164^x = 16 becomes log⁡4(4x)=log⁡4(16)\log_4(4^x) = \log_4(16), simplify to x=2x = 2
  • allows use of any logarithm base

Definition of logarithms in equations

  • log⁡b(x)=y\log_b(x) = y means by=xb^y = x (b>0b > 0, b≠1b \neq 1, x>0x > 0)
  • Rewrite logarithmic equation in exponential form
  • Solve for variable
    • log⁡3(81)=x\log_3(81) = x becomes 3x=813^x = 81, simplify to x=4x = 4
  • (base 10) often used in practical applications

One-to-one property for logarithmic equations

  • If log⁡b(x)=log⁡b(y)\log_b(x) = \log_b(y), then x=yx = y (b>0b > 0, b≠1b \neq 1, x,y>0x, y > 0)
  • Isolate logarithmic expressions on each side
  • Set arguments of logarithms equal to each other
    • log⁡(x+1)=log⁡(x−3)\log(x + 1) = \log(x - 3) becomes x+1=x−3x + 1 = x - 3, solve for x=2x = 2

Properties of Logarithms

  • Product rule: log⁡b(xy)=log⁡b(x)+log⁡b(y)\log_b(xy) = \log_b(x) + \log_b(y)
  • Quotient rule: log⁡b(x/y)=log⁡b(x)−log⁡b(y)\log_b(x/y) = \log_b(x) - \log_b(y)
  • Power rule: log⁡b(xn)=nlog⁡b(x)\log_b(x^n) = n\log_b(x)
  • These properties simplify complex logarithmic expressions

Real-world applications of exponential equations

  • and decay models:
    • Population growth (bacteria doubling), (savings accounts), (carbon dating)
    • General form: A(t)=A0[e](https://www.fiveableKeyTerm:e)ktA(t) = A_0 [e](https://www.fiveableKeyTerm:e)^{kt} (A0A_0 initial amount, kk growth/, tt time)
  • Logarithmic scales:
    • (earthquake magnitudes), (acidity/alkalinity), (sound intensity)
  • and :
    • Half-life: time for quantity to halve (medication concentration in body)
    • : time for quantity to double (cancer cell growth)
    • Used in pharmacology, radioactivity, population dynamics
  • used to describe various natural phenomena

Inverse Functions

  • Exponential and logarithmic functions are inverses of each other
  • This relationship allows solving complex equations by switching between forms

Key Terms to Review (34)

Absolute maximum: The absolute maximum of a function is the highest value that the function attains over its entire domain. It represents the peak point on the graph of the function.
Change of Base Formula: The change of base formula is a mathematical expression that allows for the conversion of logarithms from one base to another. This formula is particularly important in the context of logarithmic functions, their graphs, and the properties and equations involving logarithms and exponentials.
Common base: A common base refers to the shared base in exponential and logarithmic equations that allows for simplification and solving. It is often used to compare, combine, or solve exponential expressions by rewriting them with the same base.
Common logarithm: A common logarithm is a logarithm with base 10, often written as $\log_{10}(x)$ or simply $\log(x)$. It is commonly used in scientific calculations and when dealing with exponential growth or decay.
Common Logarithm: The common logarithm, also known as the base-10 logarithm, is a logarithmic function that expresses the power to which a base of 10 must be raised to obtain a given number. It is a fundamental concept in mathematics, with applications in various fields, including college algebra.
Compound interest: Compound interest is the interest calculated on the initial principal and also on the accumulated interest of previous periods. It is commonly used in finance and investments to calculate growth over time.
Compound Interest: Compound interest is the interest earned on interest, where the interest accrued on a principal amount is added to the original amount, and the total then earns interest in the next period. This concept is fundamental to understanding the growth of investments and loans over time.
Decay Rate: Decay rate is a measure of how quickly a quantity, such as the amount of a radioactive substance or the amplitude of a vibrating system, decreases or diminishes over time. It is a fundamental concept in the study of exponential and logarithmic equations, which describe processes that exhibit exponential growth or decay.
Decibel Scale: The decibel scale is a logarithmic unit used to measure the intensity or level of various quantities, such as sound, power, and voltage. It is commonly used in the context of acoustics, electronics, and telecommunications to quantify the relative change in a measured value.
Doubling time: Doubling time is the period it takes for a quantity to double in size or value at a constant growth rate. It is commonly used in exponential growth models.
Doubling Time: Doubling time is the amount of time it takes for a quantity to double in value. It is a crucial concept in the study of exponential growth and decay, and is closely tied to the understanding of exponential functions, their graphs, logarithmic functions, and their applications in various models.
E: e, also known as Euler's number, is a fundamental mathematical constant that is the base of the natural logarithm. It is an irrational number that is approximately equal to 2.71828 and is widely used in mathematics, science, and engineering. The term 'e' is central to the understanding of exponential functions, logarithmic functions, and their properties, which are crucial concepts in college algebra.
Exponential Decay: Exponential decay is a mathematical model that describes the gradual reduction or diminishment of a quantity over time. It is characterized by an initial value that decreases by a constant proportion during each successive time interval, resulting in an exponential decrease. This concept is fundamental to understanding various phenomena in fields such as physics, chemistry, biology, and finance.
Exponential equation: An exponential equation is an equation in which the variables appear as exponents. These equations often take the form $a^{x} = b$ where $a$ and $b$ are constants.
Exponential Equation: An exponential equation is a mathematical equation in which the unknown variable appears as the exponent of another quantity. These equations model situations where a quantity grows or decays at a constant rate over time, and are closely related to the behavior of exponential functions.
Exponential function: An exponential function is a mathematical expression in the form $f(x) = a \cdot b^x$, where $a$ is a constant, $b$ is the base greater than 0 and not equal to 1, and $x$ is the exponent. These functions model growth or decay processes.
Exponential Function: An exponential function is a mathematical function in which the independent variable appears as an exponent. These functions exhibit a characteristic curve that grows or decays at a rate proportional to the current value, leading to rapid changes in output as the input increases.
Exponential growth: Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value. This results in the function increasing rapidly over time.
Exponential Growth: Exponential growth is a pattern of change where a quantity increases at a rate proportional to its current value. This means the quantity grows by a consistent percentage over equal intervals of time, leading to rapid, accelerating growth. Exponential growth is a fundamental concept in mathematics and has applications across various fields, including biology, economics, and technology.
Exponential Model: An exponential model is a mathematical function that describes a relationship where a quantity increases or decreases at a constant rate relative to its current value. This type of model is commonly used to analyze and predict phenomena that exhibit exponential growth or decay patterns.
Growth Rate: The growth rate refers to the rate of change in a quantity over time. It is a measure of how quickly a quantity is increasing or decreasing, and is often expressed as a percentage change per unit of time. The growth rate is a crucial concept in the study of exponential functions and the analysis of exponential and logarithmic equations.
Half-life: Half-life is the time it takes for a radioactive or other substance to decay to half of its initial value. This concept is central to understanding exponential functions, their graphs, logarithmic functions, and how these models are applied to real-world situations involving growth and decay.
Inverse function: An inverse function reverses the operation of a given function. If $f(x)$ is a function, its inverse $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.
Inverse Function: An inverse function is a function that reverses the operation of another function. It undoes the original function, mapping the output back to the original input. Inverse functions are crucial in understanding the relationships between different mathematical concepts, such as domain and range, composition of functions, transformations, and exponential and logarithmic functions.
Like Bases: Like bases refer to exponential expressions that have the same base. This is an important concept in the context of solving exponential and logarithmic equations, as the properties of exponents and logarithms can be leveraged to simplify and solve these types of equations.
Logarithm: A logarithm is a mathematical function that describes the power to which a base number must be raised to get a certain value. It represents the exponent to which a base number must be raised to produce a given number. Logarithms are closely related to exponential functions and are essential in understanding topics such as logarithmic functions, graphs of logarithmic functions, exponential and logarithmic equations, and geometric sequences.
Logarithmic Function: A logarithmic function is a special type of function where the input variable is an exponent. It is the inverse of an exponential function, allowing for the determination of the exponent when the result is known. Logarithmic functions play a crucial role in various mathematical concepts and applications.
Natural logarithm: The natural logarithm is the logarithm to the base $e$, where $e$ is an irrational and transcendental number approximately equal to 2.71828. It is commonly denoted as $\ln(x)$.
Natural Logarithm: The natural logarithm, denoted as $\ln(x)$, is a logarithmic function that represents the power to which the base $e$ must be raised to get the value $x$. The natural logarithm is a fundamental concept that underpins various topics in college algebra, including logarithmic functions, their graphs, properties, and applications in solving exponential and logarithmic equations, as well as modeling real-world phenomena.
One-to-One Property: The one-to-one property is a fundamental characteristic of certain functions, where each element in the domain is paired with a unique element in the range. This property is crucial in the context of exponential and logarithmic equations, as it ensures a direct and unambiguous relationship between the input and output values.
PH Scale: The pH scale is a measure of the acidity or basicity of a solution, ranging from 0 to 14. It is a logarithmic scale that indicates the concentration of hydrogen ions (H+) in a solution, with lower values representing more acidic solutions and higher values representing more basic (alkaline) solutions.
Properties of Logarithms: The properties of logarithms are the fundamental rules that govern the behavior of logarithmic functions. These properties describe how logarithms can be manipulated and used to solve exponential and logarithmic equations, which are the focus of the 6.6 Exponential and Logarithmic Equations topic.
Radioactive decay: Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This process is a key example of exponential decay, where the amount of radioactive substance decreases over time at a rate proportional to its current amount. Understanding radioactive decay is crucial for applications in fields like nuclear physics, radiometric dating, and medical imaging.
Richter Scale: The Richter scale is a logarithmic scale used to measure the magnitude of earthquakes. It was developed in 1935 by American seismologist Charles Richter and is a fundamental tool in understanding the strength and impact of seismic events.
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Glossary
Glossary
6.7 Exponential and Logarithmic Models