--- title: Logarithm Rules 1 - Basic Properties, Formulas & Examples description: "Logarithm rules, part 1: product, quotient, and power rules for simplifying log expressions." url: https://astra-ai.co/videos/logarithms/rules-for-calculating-logarithms/logarithm-rules-1/ locale: en type: video-lesson keywords: - logarithm rules - product rule for logs - quotient rule for logs - power rule published: 2026-10-01T09:56:23.429Z updated: 2026-10-04T22:00:28.265Z alternates: fr: https://astra-ai.co/fr/videos/logarithmes/regles-de-calcul-des-logarithmes/regles-des-logarithmes-1/ en: https://astra-ai.co/videos/logarithms/rules-for-calculating-logarithms/logarithm-rules-1/ de: https://astra-ai.co/de/videos/logarithmen/regeln-fur-die-berechnung-von-logarithmen/logarithmische-regeln-1/ it: https://astra-ai.co/it/videos/logaritmi/regole-calcolare-logaritmi/regole-logaritmi-1/ pl: https://astra-ai.co/pl/videos/logarytmy/wzory-obliczania-logarytmow/wzory-logarytmiczne-1/ sl: https://astra-ai.co/sl/videos/logaritmi/pravila-racunanja-z-logaritmi/pravila-logaritmov-1/ hr: https://astra-ai.co/hr/videos/logaritmi/pravila-racunanja-s-logaritmima/pravila-logaritama-1/ pt: https://astra-ai.co/pt/videos/logaritmos/regras-de-calculo-com-logaritmos/regras-para-logaritmos-1/ fi: https://astra-ai.co/fi/videos/logaritmit/logaritmien-laskusaannot/logaritmien-saannot-1/ nl: https://astra-ai.co/nl/videos/logaritmen/rekenregels-met-logaritmen/regels-voor-logaritmen-1/ da: https://astra-ai.co/da/videos/logaritmer/regneregler-for-logaritmer/logaritmeregler-1/ el: https://astra-ai.co/el/videos/logarithmoi/kanones-ypologismou-logarithmon/logarithm-rules-1/ no: https://astra-ai.co/no/videos/logaritmer/regler-for-beregning-av-logaritmer/logaritmeregler-1/ cs: https://astra-ai.co/cs/videos/logaritmy/pravidla-vypoctu-s-logaritmy/pravidla-logaritmu-1/ bg: https://astra-ai.co/bg/videos/logaritmi/pravila-za-presmyatane-s-logaritmi/pravila-za-logaritmi-1/ ro: https://astra-ai.co/ro/videos/logaritmi/reguli-de-calcul-al-logaritmilor/lectie-video-5f32b3e0/ id: https://astra-ai.co/id/videos/logaritma/aturan-menghitung-logaritma/aturan-logaritma-1/ sv: https://astra-ai.co/sv/videos/logaritmer/regler-for-berakning-med-logaritmer/logaritmregler-1/ hu: https://astra-ai.co/hu/videos/logaritmusok/logaritmusszamitas-szabalyai/logaritmusszabalyok-1/ mx: https://astra-ai.co/mx/videos/logaritmos/capitulo-6/leccion-12/ vi: https://astra-ai.co/vi/videos/vi-topic-17/vi-chapter-10/vi-logarithm-rules-1/ uk: https://astra-ai.co/uk/videos/uk-topic-17/uk-chapter-10/uk-logarithm-rules-1/ lt: https://astra-ai.co/lt/videos/logaritmai/logaritmu-skaiciavimo-taisykles/pamoka-5f32b3e0/ lv: https://astra-ai.co/lv/videos/logaritmi/logaritmu-aprekinasanas-likumi/stunda-5f32b3e0/ sk: https://astra-ai.co/sk/videos/logaritmy/kapitola-006/lekcia-012/ tr: https://astra-ai.co/tr/videos/logaritmalar/bolum-006/ders-012/ sr: https://astra-ai.co/sr/videos/logaritmi/poglavlje-006/lekcija-012/ th: https://astra-ai.co/th/videos/lokarithuem/bot-10-kot-kan-khanan-lokarithuem/botrian-012/ ms: https://astra-ai.co/ms/videos/logaritma/bab-10-matematik-asas/pelajaran-012/ my: https://astra-ai.co/my/videos/my-d4a82fdb/my-1181c2a2/lesson-5f32b3e0/ lo: https://astra-ai.co/lo/videos/lo-d4a82fdb/lo-1181c2a2/lesson-5f32b3e0/ km: https://astra-ai.co/km/videos/logarithms/chapter-1181c2a2/lesson-5f32b3e0/ zh-TW: https://astra-ai.co/zh-TW/videos/logarithms/chapter-1181c2a2/lesson-5f32b3e0/ es-AR: https://astra-ai.co/es-AR/videos/logaritmos/capitulo-6/leccion-12/ zh-HK: https://astra-ai.co/zh-HK/videos/shuxue-kecheng-017/shuxue-zhang-010/shuxue-kecheng-012/ br: https://astra-ai.co/br/videos/logaritmos/capitulo-matematica-010/video-matematica-012/ en-PH: https://astra-ai.co/en-PH/videos/logarithms/chapter-1181c2a2/lesson-5f32b3e0/ en-IN: https://astra-ai.co/en-IN/videos/logarithms/chapter-1181c2a2/logarithm-rules-1/ es-PE: https://astra-ai.co/es-PE/videos/logaritmos/capitulo-video-1181c2a2/leccion-de-video-5f32b3e0/ ms-BN: https://astra-ai.co/ms-BN/videos/logaritma/chapter-1181c2a2/pelajaran-5f32b3e0/ pt-TL: https://astra-ai.co/pt-TL/videos/logaritmos/chapter-1181c2a2/licao-de-video-012/ et: https://astra-ai.co/et/videos/logaritmid/logaritmide-arvutamise-reeglid/logaritmide-reeglid-1/ es-CL: https://astra-ai.co/es-CL/videos/logaritmos/capitulo-6/reglas-de-los-logaritmos-1/ ko: https://astra-ai.co/ko/videos/logarithms-ko/06/012/ ja: https://astra-ai.co/ja/videos/logarithms-ja/ja-chapter-06/012/ es: https://astra-ai.co/es/videos/logaritmos/reglas-de-los-logaritmos/reglas-de-los-logaritmos-1/ --- # Logarithm Rules 1 > Logarithm rules, part 1, cover the product, quotient, and power rules. This lesson uses them to simplify log expressions. ## INTRODUCTION TO LOGARITHMS AND THEIR RULES Logarithms are mathematical tools used to solve equations where the unknown is in the exponent. The fundamental definition of a logarithm states that log\_b(x) = y if and only if b^y = x, where 'b' is a positive number not equal to 1, and 'x' is a positive real number. To facilitate working with logarithms, there are established rules derived from the laws of exponentiation that allow logarithmic expressions to be simplified and reformulated. ## BASIC RULES FOR LOGARITHMS ### PRODUCT RULE The logarithm of the product of two numbers is equal to the sum of the logarithms of the individual factors: log\_b(x \* y) = log\_b(x) + log\_b(y) Example: log₂(8) = log₂(4 \* 2) = log₂(4) + log₂(2) = 2 + 1 = 3 ### QUOTIENT RULE The logarithm of the quotient of two numbers is equal to the difference of their logarithms: log\_b(x / y) = log\_b(x) - log\_b(y) Example: log₂(8 / 4) = log₂(8) - log₂(4) = 3 - 2 = 1 ### POWER RULE The logarithm of a power is equal to the product of the exponent and the logarithm of the base of the power: log\_b(xⁿ) = n \* log\_b(x) Example: log₂(8) = log₂(2³) = 3 \* log₂(2) = 3 \* 1 = 3 ### CHANGE OF BASE RULE The logarithm to base 'b' can be expressed using the logarithm of any other base 'c': log\_b(x) = log\_c(x) / log\_c(b) Example: log₂(8) = log₁₀(8) / log₁₀(2) ≈ 0.903 / 0.301 ≈ 3 ### LOGARITHM OF UNITY AND LOGARITHM OF THE BASE - log\_b(1) = 0, since b⁰ = 1 - log\_b(b) = 1, since b¹ = b Examples: log₂(1) = 0 log₂(2) = 1 ## RELATIONSHIP WITH EXPONENTIAL FUNCTIONS The logarithm is the inverse function of the exponential function. This means: If b^y = x, then y = log\_b(x). Example for solving an equation: If we have 2^x = 8, then by taking the logarithm of both sides (to base 2): x = log₂(8) = 3 ## CONCLUSION The rules for logarithms are fundamental for understanding and working with logarithms. They enable us to: - Simplify complex expressions - Solve logarithmic equations - Convert between different logarithm bases - Work accurately in algebra and analysis Knowledge of these rules provides us with a more powerful tool for mastering exponential expressions and a broader mathematical understanding.