Algebra 2 Practice - Solving Logarithmic Equations (Example 3)

Please subscribe!    / nickperich   *Solving Logarithmic Equations* in Algebra 2 involves isolating the logarithmic expression and applying properties of logarithms and exponents to find the solution. These equations often require understanding the relationship between logarithms and exponents, as well as applying the logarithmic properties effectively. --- General Steps to Solve Logarithmic Equations: 1. **Isolate the Logarithmic Term**: Rearrange the equation so that a single logarithmic term is isolated on one side. 2. **Apply Logarithmic or Exponential Properties**: Use the definition of a logarithm: \(\log_b(x) = y \iff b^y = x\). Use properties of logarithms, such as: Product Property: \(\log_b(MN) = \log_b(M) + \log_b(N)\) Quotient Property: \(\log_b(M/N) = \log_b(M) - \log_b(N)\) Power Property: \(\log_b(M^p) = p \cdot \log_b(M)\) 3. **Rewrite in Exponential Form (if needed)**: If a single logarithm equals a value (e.g., \(\log_b(x) = y\)), rewrite it in exponential form to solve. 4. **Solve for the Variable**: Use algebraic methods to solve the resulting equation. 5. **Check for Extraneous Solutions**: Logarithms are undefined for negative numbers and zero, so ensure all solutions satisfy the domain of the original equation. --- Example Problem 1: Solve \(\log_2(x) = 5\). #### Solution: 1. Rewrite in exponential form: \(2^5 = x\). 2. Simplify: \(x = 32\). --- Example Problem 2: Solve \(\log_3(x + 4) = 2\). #### Solution: 1. Rewrite in exponential form: \(3^2 = x + 4\). 2. Simplify: \(9 = x + 4\). 3. Solve for \(x\): \(x = 5\). --- Example Problem 3: Solve \(\log_2(x) + \log_2(x - 3) = 3\). #### Solution: 1. Apply the Product Property: \(\log_2(x(x - 3)) = 3\). 2. Simplify: \(\log_2(x^2 - 3x) = 3\). 3. Rewrite in exponential form: \(2^3 = x^2 - 3x\). 4. Simplify: \(8 = x^2 - 3x\). 5. Rearrange into standard quadratic form: \(x^2 - 3x - 8 = 0\). 6. Solve using factoring or the quadratic formula: \((x - 4)(x + 2) = 0\). 7. Solutions: \(x = 4\) or \(x = -2\). 8. Check for extraneous solutions: Logarithms are undefined for \(x \leq 0\). Valid solution: \(x = 4\). --- Example Problem 4: Solve \(2\log_5(x) = \log_5(16)\). #### Solution: 1. Simplify using the Power Property: \(\log_5(x^2) = \log_5(16)\). 2. Since the bases and arguments are equal, set the arguments equal: \(x^2 = 16\). 3. Solve for \(x\): \(x = \pm4\). 4. Check for extraneous solutions: \(x = -4\) is invalid (logarithms cannot take negative arguments). Valid solution: \(x = 4\). --- Tips for Practice: Always check the domain of logarithmic expressions. Simplify using logarithmic properties before converting to exponential form. Look out for extraneous solutions when squaring or solving quadratics. Would you like me to create practice problems for this topic? 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