Logarithm - 1 Comparing the range of constant logarithm (using power law)
- Logarithms are the inverse of exponentiation. - They help solve exponential equations by reducing them to linear equations. - Logarithms have various applications in exponential growth and decay problems. - In this lecture, we will focus on comparing the range of constant logarithms using the power law. - Let's dive into the topic!
Power Rule of Logarithms: For any positive numbers a, b, and x, and any positive integer n: **log<sub>b</sub>(a<sup>n</sup>x) = n*log<sub>b</sub>(a) + log<sub>b</sub>(x)** This rule helps us manipulate logarithmic expressions and simplify them.
Example 1: Simplify the logarithmic expression **log<sub>2</sub>(4x<sup>2</sup>)** using the power rule. Solution: Using the power rule, we have: log<sub>2</sub>(4x<sup>2</sup>) = 2*log<sub>2</sub>(2) + log<sub>2</sub>(x<sup>2</sup>) Simplifying further: = 2*1 + 2*log<sub>2</sub>(x) = 2 + 2*log<sub>2</sub>(x) Therefore, log<sub>2</sub>(4x<sup>2</sup>) simplifies to 2 + 2*log<sub>2</sub>(x).
Power Law of Logarithms: The power law states that for any positive numbers a, b, and c: **log<sub>b</sub>(a<sup>c</sup>) = c*log<sub>b</sub>(a)** This rule helps us simplify logarithmic equations involving exponents.
Example 2: Simplify the logarithmic equation **log<sub>5</sub>(125<sup>2</sup>) = x** using the power law. Solution: Using the power law, we have: log<sub>5</sub>(125<sup>2</sup>) = 2*log<sub>5</sub>(125) Since 125 = 5<sup>3</sup>, we can simplify further: = 2*log<sub>5</sub>(5<sup>3</sup>) = 2*3 = 6 Therefore, the logarithmic equation log<sub>5</sub>(125<sup>2</sup>) = x simplifies to x = 6.
Properties of Logarithms: - log<sub>b</sub>(xy) = log<sub>b</sub>(x) + log<sub>b</sub>(y) (Product Rule) - log<sub>b</sub>(x/y) = log<sub>b</sub>(x) - log<sub>b</sub>(y) (Quotient Rule) - log<sub>b</sub>(x<sup>n</sup>) = n*log<sub>b</sub>(x) (Power Rule) - log<sub>b</sub>(1/x) = -log<sub>b</sub>(x) (Reciprocal Rule) - log<sub>b</sub>(b) = 1 (Identity Rule) - log<sub>b</sub>(1) = 0 (Zero Rule)
Example 3: Simplify the logarithmic expression **log<sub>3</sub>(9x) - log<sub>3</sub>(y<sup>2</sup>)** using the properties of logarithms. Solution: Using the Quotient Rule of logarithms, we have: log<sub>3</sub>(9x) - log<sub>3</sub>(y<sup>2</sup>) = log<sub>3</sub>(9x/y<sup>2</sup>) Now, we can simplify further using the Product Rule: = log<sub>3</sub>(9) + log<sub>3</sub>(x) - log<sub>3</sub>(y<sup>2</sup>) = 2 + log<sub>3</sub>(x) - 2*log<sub>3</sub>(y) Therefore, the logarithmic expression log<sub>3</sub>(9x) - log<sub>3</sub>(y<sup>2</sup>) simplifies to 2 + log<sub>3</sub>(x) - 2*log<sub>3</sub>(y).
Inverse Properties of Exponents and Logarithms: - a<sup>log<sub>a</sub>(x)</sup> = x - log<sub>a</sub>(a<sup>x</sup>) = x These properties highlight the relationship between exponents and logarithms as inverse operations.
Example 4: Solve the exponential equation **2<sup>3x - 1</sup> = 8** using inverse properties of exponents and logarithms. Solution: Using the inverse property of exponents and logarithms, we can rewrite the equation as: </section> <section style="font-size: 50px";> <p>3x - 1 = log<sub>2</sub>(8) Since 8 = 2<sup>3</sup>, we have:</p> </section> <section style="font-size: 50px";> <p>3x - 1 = 3 Simplifying further:</p> </section> <section style="font-size: 50px";> <p>3x = 4 x = 4/3 Therefore, the solution to the exponential equation 2<sup>3x - 1</sup> = 8 is x = 4/3.
Logarithm - 1 Comparing the range of constant logarithm (using power law)
- Logarithms are the inverse of exponentiation. - They help solve exponential equations by reducing them to linear equations. - Logarithms have various applications in exponential growth and decay problems. - In this lecture, we will focus on comparing the range of constant logarithms using the power law. - Let's dive into the topic!
Power Rule of Logarithms: For any positive numbers a, b, and x, and any positive integer n: **log_b(a^n * x) = n * log_b(a) + log_b(x)** This rule helps us manipulate logarithmic expressions and simplify them.
Example 1: Simplify the logarithmic expression **log_2(4x^2)** using the power rule. Solution: Using the power rule, we have: log_2(4x^2) = 2 * log_2(2) + log_2(x^2) Simplifying further: = 2 * 1 + 2 * log_2(x) = 2 + 2 * log_2(x) Therefore, log_2(4x^2) simplifies to 2 + 2 * log_2(x).
Power Law of Logarithms: The power law states that for any positive numbers a, b, and c: **log_b(a^c) = c * log_b(a)** This rule helps us simplify logarithmic equations involving exponents.
Example 2: Simplify the logarithmic equation **log_5(125^2) = x** using the power law. Solution: Using the power law, we have: log_5(125^2) = 2 * log_5(125) Since 125 = 5^3, we can simplify further: = 2 * log_5(5^3) = 2 * 3 = 6 Therefore, the logarithmic equation log_5(125^2) = x simplifies to x = 6.
Properties of Logarithms: - log_b(xy) = log_b(x) + log_b(y) (Product Rule) - log_b(x/y) = log_b(x) - log_b(y) (Quotient Rule) - log_b(x^n) = n * log_b(x) (Power Rule) - log_b(1/x) = -log_b(x) (Reciprocal Rule) - log_b(b) = 1 (Identity Rule) - log_b(1) = 0 (Zero Rule)
Example 3: Simplify the logarithmic expression **log_3(9x) - log_3(y^2)** using the properties of logarithms. Solution: Using the Quotient Rule of logarithms, we have: log_3(9x) - log_3(y^2) = log_3(9x/y^2) Now, we can simplify further using the Product Rule: = log_3(9) + log_3(x) - log_3(y^2) = 2 + log_3(x) - 2 * log_3(y) Therefore, the logarithmic expression log_3(9x) - log_3(y^2) simplifies to 2 + log_3(x) - 2 * log_3(y).
Inverse Properties of Exponents and Logarithms: - a^(log_a(x)) = x - log_a(a^x) = x These properties highlight the relationship between exponents and logarithms as inverse operations.
Example 4: Solve the exponential equation **2^(3x - 1) = 8** using inverse properties of exponents and logarithms. Solution: Using the inverse property of exponents and logarithms, we can rewrite the equation as: </section> <section style="font-size: 50px";> <p>3x - 1 = log_2(8) Since 8 = 2^3, we have:</p> </section> <section style="font-size: 50px";> <p>3x - 1 = 3 Simplifying further:</p> </section> <section style="font-size: 50px";> <p>3x = 4 x = 4/3 Therefore, the solution to the exponential equation 2^(3x - 1) = 8 is x = 4/3.
Exponential and Logarithmic Functions: - Exponential functions: functions of the form y = a^x, where a is a positive constant. - Logarithmic functions: functions of the form y = log_b(x), where b is a positive constant. - Exponential and logarithmic functions are inversely related. - They are used to model real-life situations involving growth, decay, and rates of change. - Let's explore some examples to understand these functions better.
Example 1: A bacteria population doubles every 12 hours. Write an exponential function to represent the population at time t. Solution: Since the population is doubling, the base of the exponential function is 2. The time period is 12 hours, so the function becomes: P(t) = 2^(t/12) Here, P(t) represents the population at time t.
Comparing the Growth Rates: - Exponential growth: occurs when the base a > 1. - The function increases rapidly as x increases. - Logarithmic growth: occurs when the base 0 < a < 1. - The function increases slowly as x increases. - The nature of the base affects the growth rate and behavior of the function.
Example 2: Compare the growth rates of the exponential functions y = 2^x and y = (1/2)^x. Solution: For y = 2^x, as x increases, the function grows rapidly. For y = (1/2)^x, as x increases, the function grows slowly. Therefore, the first function represents exponential growth, while the second function represents logarithmic growth.
Graphical Representation: - Exponential functions have a curved, J-shaped graph. - Logarithmic functions have a curved, S-shaped graph. - The intercepts, shape, and behavior of the graph depend on the base and exponent of the function. - Let's visualize this with some examples.
Example 3: Graph the exponential function y = 3^x and the logarithmic function y = log_3(x). Solution: The exponential function y = 3^x has a J-shaped graph, increasing rapidly as x increases. The logarithmic function y = log_3(x) has an S-shaped graph, increasing slowly as x increases. The intercepts and other properties of the graph can be determined using the properties of exponential and logarithmic functions.
Domain and Range: - The domain of an exponential function is all real numbers. - The range of an exponential function depends on the base and direction of growth. - The domain of a logarithmic function is positive real numbers. - The range of a logarithmic function is all real numbers. - Let's explore some examples to understand this better.
Example 4: Find the domain and range of the exponential function y = 2^x. Solution: The domain of y = 2^x is all real numbers since x can take any value. The range depends on the direction of growth, which is positive. Therefore, the range is all positive real numbers. Hence, the domain is (-∞, +∞) and the range is (0, +∞).
Applications of Logarithms: - Logarithms are used to calculate exponential growth and decay rates. - They are used in finance, biology, chemistry, and </section> </div> </div> <script src="/reveal/dist/reveal.js"></script> <script src="/reveal/plugin/markdown/markdown.js"></script> <script src="/reveal/plugin/highlight/highlight.js"></script> <script src="/reveal/plugin/math/math.js"></script> <script src="/reveal/plugin/anything/plugin.js"></script> <script src="/reveal/plugin/notes/notes.js"></script> <script src="/reveal/plugin/chart/Chart.min.js"></script> <script src="/reveal/plugin/chart/plugin.js"></script> <script src="/reveal/plugin/menu/menu.js"></script> <script src="/reveal/plugin/highlight/highlight.js"></script> <script src="/reveal/plugin/audio-slideshow/plugin.js"></script> <script src="/reveal/plugin/audio-slideshow/recorder.js"></script> <script src="/reveal/plugin/audio-slideshow/RecordRTC.js"></script> <script src="/reveal/plugin/copycode/copycode.js"></script> <script src="/reveal/plugin/RevealEditor-master/revealeditor.js"></script> <script src="/reveal/plugin/jump/jump.js"></script> <script src="/reveal/plugin/zoom/zoom.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/clipboard.js/2.0.6/clipboard.min.js"></script> <script> Reveal.initialize({ slideNumber: true, controls: true, controlsTutorial: true, progress: true, transition: 'fade', highlight: { escapeHTML: false }, customcontrols: { controls: [ { icon: '<i class="fa fa-pen-square"></i>', title: 'Toggle chalkboard (B)', action: 'RevealChalkboard.toggleChalkboard();' }, { icon: '<i class="fa fa-pen"></i>', title: 'Toggle notes canvas (C)', action: 'RevealChalkboard.toggleNotesCanvas();' } ] }, plugins: [RevealMath.KaTeX,RevealMarkdown, RevealHighlight, RevealNotes, RevealAnything, RevealMenu, RevealChalkboard, RevealChart, RevealAudioSlideshow, RevealAudioRecorder, CopyCode, RevealZoom], }); 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