https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:exponential-growth-decay/x2f8bb11595b61c86:exponential-vs-linear-growth/v/exponential-growth-functions 2022-11-16T04:22:43.7414897Z Intro to exponential functions An exponential function represents the relationship between an input and output, where we use repeated multiplication on an initial value to get the output for any given input. Exponential functions can grow or decay very quickly. Exponential functions are often used to model things in the real world, such as populations, radioactive materials, and compound interest. Sal Khan video https://cdn.kastatic.org/googleusercontent/MOdCeqFGlDqk69XEhUqpMPoJA42wLRt7GwZjEAthvS_AxHPH0ieVuA4DqeI3ImMs5u7K5aCZdV6MCqKxXrs-d4g Intro to exponential functions An exponential function represents the relationship between an input and output, where we use repeated multiplication on an initial value to get the output for any given input. Exponential functions can grow or decay very quickly. Exponential functions are often used to model things in the real world, such as populations, radioactive materials, and compound interest. https://cdn.kastatic.org/ka-youtube-converted/6WMZ7J0wwMI.mp4/6WMZ7J0wwMI.mp4 461 Exponential and logarithmic functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:exponential-growth-decay/x2f8bb11595b61c86:exponential-decay/v/exponential-decay-intro 2022-11-16T04:22:43.7414897Z Exponential decay intro Both exponential growth and decay functions involve repeated multiplication by a constant factor. However, the difference lies in the size of that factor: - In an exponential growth function, the factor is greater than 1, so the output will increase (or "grow") over time. - In an exponential decay function, the factor is between 0 and 1, so the output will decrease (or "decay") over time. video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Exponential decay intro Both exponential growth and decay functions involve repeated multiplication by a constant factor. However, the difference lies in the size of that factor: - In an exponential growth function, the factor is greater than 1, so the output will increase (or "grow") over time. - In an exponential decay function, the factor is between 0 and 1, so the output will decrease (or "decay") over time. https://cdn.kastatic.org/ka-youtube-converted/v4IdaXvyE7U.mp4/v4IdaXvyE7U.mp4 492 Exponential and logarithmic functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:exponential-growth-decay/x2f8bb11595b61c86:graphs-of-exponential-growth/v/graphs-of-exponential-growth 2021-02-16T05:02:26Z Graphs of exponential growth Identifying which graph represents a given exponential function. video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Graphs of exponential growth Identifying which graph represents a given exponential function. https://cdn.kastatic.org/ka-youtube-converted/6rX2VNybXEE.mp4/6rX2VNybXEE.mp4 261 Exponential and logarithmic functions https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:exponential-growth-decay/x2f8bb11595b61c86:exponential-decay/e/exponential-growth-vs-decay 2025-03-07T16:29:07.940270514Z Exponential growth vs. decay Identify whether an exponential functions represents growth or decay. Tomer Gal exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:exponential-growth-decay/x2f8bb11595b61c86:graphs-of-exponential-growth/e/graphing-exponential-growth-intro 2025-03-03T19:47:54.930712342Z Graphs of exponential growth Start exploring the relationship between exponential functions and their graphs. Tomer Gal exercise https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:transformations/x2ec2f6f830c9fb89:exp-graphs/e/graphs-of-exponential-functions 2026-04-07T22:59:30.107127435Z Graphs of exponential functions Graph exponential functions and find the appropriate graph given the function. Tomer Gal exercise https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:log-intro/v/logarithms 2021-06-30T17:32:37Z Intro to logarithms Sal explains what logarithms are and gives a few examples of finding logarithms. Sal Khan video https://cdn.kastatic.org/googleusercontent/_RkZoOHKtmVk5F8_teMl8HJnKuYpGcFOUu9QgleddEiix-1mZsZf4T2CRlqmZLeq2jmOuybi1N6RCIMjkLHMTO8png Intro to logarithms Sal explains what logarithms are and gives a few examples of finding logarithms. https://cdn.kastatic.org/ka-youtube-converted/Z5myJ8dg_rM.mp4/Z5myJ8dg_rM.mp4 422 Exponential and logarithmic functions https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:log-prop/v/introduction-to-logarithm-properties 2021-06-30T17:32:40Z Intro to logarithm properties (1 of 2) Sal introduces the logarithm identities for addition and subtraction of logarithms. Sal Khan video https://cdn.kastatic.org/googleusercontent/nzxy-IVnxWaW853loCOsr_TO6wouI3i0a4bHGL0q5bb_5-zRXhbax5LHz-9nuvkjyGKSZNxiidIs2_1ODvT7ocFl Intro to logarithm properties (1 of 2) Sal introduces the logarithm identities for addition and subtraction of logarithms. https://cdn.kastatic.org/ka-youtube-converted/PupNgv49_WY.mp4/PupNgv49_WY.mp4 556 Exponential and logarithmic functions https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:log-prop/v/introduction-to-logarithm-properties-part-2 2021-06-30T17:32:38Z Intro to logarithm properties (2 of 2) Sal introduces the logarithm identities for multiplication of logarithm by a constant, and the change of base rule. Sal Khan video https://cdn.kastatic.org/googleusercontent/oldCPdfzyfwKtz0lxMji9ZEt-Nau-FEnKaRQWr6BpDyEgcuA17Km-UJApBrtIFhWDW8N6ou6moT-zWtsO7Q0Jcek Intro to logarithm properties (2 of 2) Sal introduces the logarithm identities for multiplication of logarithm by a constant, and the change of base rule. https://cdn.kastatic.org/ka-youtube-converted/TMmxKZaCqe0.mp4/TMmxKZaCqe0.mp4 604 Exponential and logarithmic functions https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:log-intro/e/understanding-logs-as-inverse-exponentials 2026-04-07T22:23:57.656416987Z Relationship between exponentials & logarithms Solve various problems that focus on the relationship between a^x=b and log_a(b)=x. Stacy Beaudoin exercise https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:transformations/x2ec2f6f830c9fb89:log-graphs/e/graphs-of-exponentials-and-logarithms 2026-04-14T11:30:38.220334671Z Graphs of logarithmic functions Graph logarithmic functions and find the appropriate graph given the function. Tomer Gal exercise https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:log-prop/e/logarithms_2 2025-12-22T06:46:20.288709789Z Use the properties of logarithms Use the properties of logarithms in order to rewrite a given expression in an equivalent, different form. Rob Yates exercise https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:change-of-base/e/rewrite-logarithmic-expressions-using-the-change-of-base-rule 2025-05-05T08:14:25.350149109Z Use the logarithm change of base rule Find the logarithmic variable expression that is equivalent to a given logarithmic expression. Stacy Beaudoin exercise https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-7/v/derivative-of-ex 2024-08-20T23:45:10.987222765Z Derivative of 𝑒ˣ Discover the fascinating property of the derivative of 𝑒ˣ, which remains 𝑒ˣ after differentiation. This unique characteristic is central to our understanding and exploration of exponential functions, showcasing the remarkable nature of the number 𝑒. video https://cdn.kastatic.org/ka_thumbnails_cache/394d681d-e39a-4cd2-90fa-26c4a7aa1e6e_1280_720_base.png Derivative of 𝑒ˣ Discover the fascinating property of the derivative of 𝑒ˣ, which remains 𝑒ˣ after differentiation. This unique characteristic is central to our understanding and exploration of exponential functions, showcasing the remarkable nature of the number 𝑒. https://cdn.kastatic.org/ka-youtube-converted/W_gNAjWWvBg.mp4/W_gNAjWWvBg.mp4 182 Exponential and logarithmic functions https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab-3-1b/v/logarithmic-functions-differentiation-intro 2023-07-27T02:34:53.22139958Z Derivative of logₐx (for any positive base a≠1) Now let's look into the fascinating world of logarithms, exploring how to find the derivative of logₐx for any positive base a≠1. Leveraging the derivative of ln(x) and the change of base rule, we successfully differentiate log₇x and -3log_π(x). Join us on this mathematical journey! video https://cdn.kastatic.org/ka_thumbnails_cache/5b780056-99a3-422f-8206-61ea0fd486af_1280_720_base.png Derivative of logₐx (for any positive base a≠1) Now let's look into the fascinating world of logarithms, exploring how to find the derivative of logₐx for any positive base a≠1. Leveraging the derivative of ln(x) and the change of base rule, we successfully differentiate log₇x and -3log_π(x). Join us on this mathematical journey! https://cdn.kastatic.org/ka-youtube-converted/2XB4K6T-yKo.mp4/2XB4K6T-yKo.mp4 288 Exponential and logarithmic functions https://www.khanacademy.org/math/revision-term-1-ncert-math-class-12/x6a634f0b79812b7f:week-2/x6a634f0b79812b7f:continuity-and-differentiability/e/working-on-exponential-and-logarithmic-functions 2026-04-08T00:54:09.631945657Z Working on exponential and logarithmic functions The exercise has problems from, Test yourself: exponential and logarithmic functions. And the exercise will be do-7. Devashish Phadnis exercise