Solving Logarithmic Equations
Solving logarithmic equations is a Grade 11 algebra skill in Big Ideas Math. Key strategies include using the definition of logarithm (logₐ(x) = y means aʸ = x) to convert to exponential form, and applying logarithm properties: product rule (log(AB) = log A + log B), quotient rule (log(A/B) = log A − log B), and power rule (log(Aⁿ) = n·log A). To solve log₂(x) = 5: convert to 2⁵ = x, giving x = 32. Always check solutions in the original equation because logarithms are undefined for non-positive arguments—extraneous solutions must be rejected.
Key Concepts
To solve logarithmic equations, change the equation to exponential form and solve. If $\log b x = y$, then $b^y = x$. Always check solutions to eliminate any that make the logarithm's argument non positive.
Common Questions
How do you convert a logarithmic equation to exponential form?
Use the definition: logₐ(x) = y is equivalent to aʸ = x. For log₂(x) = 5: 2⁵ = x, so x = 32.
What are the three key logarithm properties for solving equations?
Product rule: log(AB) = logA + logB. Quotient rule: log(A/B) = logA − logB. Power rule: log(Aⁿ) = n·logA.
How do you solve log₃(2x − 1) = 2?
Convert to exponential form: 3² = 2x − 1 → 9 = 2x − 1 → 2x = 10 → x = 5. Check: 2(5)−1 = 9 > 0, so x = 5 is valid.
What are extraneous solutions in logarithmic equations?
Extraneous solutions satisfy the solved equation algebraically but make the original logarithm's argument zero or negative (undefined). Always check solutions in the original equation.
How do you solve an equation with logarithms on both sides?
If logₐ(A) = logₐ(B), then A = B. Set the arguments equal and solve the resulting equation. Then verify both arguments are positive.
What is the change-of-base formula and when is it used?
log_b(x) = log(x)/log(b) (or ln(x)/ln(b)). It converts logarithms of any base to common or natural log for calculator evaluation.