Algebra 2 Practice - Simply an Exponential Function Using Logarithm Rules (Example 1)

Please subscribe!    / nickperich   To simplify an exponential function using logarithm rules, we use properties of logarithms to transform and simplify the expression. Here are the key logarithmic properties: 1. *Power Rule:* \[ \log_b(x^n) = n \cdot \log_b(x) \] This rule allows you to bring the exponent down as a multiplier. 2. *Product Rule:* \[ \log_b(x \cdot y) = \log_b(x) + \log_b(y) \] This rule allows you to break up the logarithm of a product into the sum of two logarithms. 3. *Quotient Rule:* \[ \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \] This rule allows you to break up the logarithm of a quotient into the difference of two logarithms. 4. *Change of Base Formula:* \[ \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \] This formula allows you to change the base of a logarithm to any new base \(k\), typically 10 or \(e\), to make calculations easier. Example 1: Simplifying Using the Power Rule Simplify the following expression: \[ \log_2(8^4) \] By applying the **Power Rule**: \[ \log_2(8^4) = 4 \cdot \log_2(8) \] Since \(8 = 2^3\), \(\log_2(8) = 3\), so: \[ 4 \cdot \log_2(8) = 4 \cdot 3 = 12 \] Thus, \(\log_2(8^4) = 12\). Example 2: Simplifying Using the Product Rule Simplify the following expression: \[ \log_5(25) + \log_5(4) \] By applying the **Product Rule**: \[ \log_5(25) + \log_5(4) = \log_5(25 \cdot 4) = \log_5(100) \] Since \(25 = 5^2\), we know that \(\log_5(25) = 2\), so: \[ \log_5(25) + \log_5(4) = \log_5(100) = 2 \] Thus, the expression simplifies to 2. Example 3: Simplifying Using the Quotient Rule Simplify the following expression: \[ \log_3\left(\frac{81}{9}\right) \] By applying the **Quotient Rule**: \[ \log_3\left(\frac{81}{9}\right) = \log_3(81) - \log_3(9) \] Since \(81 = 3^4\) and \(9 = 3^2\), we have: \[ \log_3(81) = 4 \quad \text{and} \quad \log_3(9) = 2 \] So: \[ \log_3\left(\frac{81}{9}\right) = 4 - 2 = 2 \] Thus, the expression simplifies to 2. Would you like more practice examples or further explanation on how to apply these rules? I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out: / nickperich Nick Perich Norristown Area High School Norristown Area School District Norristown, Pa #math #algebra #algebra2 #maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study