Solving advanced logarithmic equations
Solving Logarithmic Equations
Solving Logarithmic Equations is a topic in Exponential & Logarithmic Functions in the California Common Core State Standards. It is aligned to Standard F-LE.4, which requires students to solve logarithmic equations and identify extraneous solutions.
Solving a logarithmic equation condenses to a single logarithm, rewrites in exponential form, and checks each solution against the domain to reject extraneous roots.
Theory
A logarithmic equation has the unknown inside a logarithm. The reliable plan:
- Condense to a single logarithm using the log properties.
- Rewrite in exponential form \(\log_b x=c\Rightarrow x=b^c\).
- Solve the resulting equation.
- Check the domain — reject any candidate that makes the input of a log zero or negative.
The key conversion and the domain requirement:
How to solve a logarithmic equation
- Combine all logs into a single logarithm.
- Convert to exponential form.
- Solve the algebraic equation.
- Verify each solution keeps every log's argument positive.
Rewrite in exponential form.
| \(x\) | \(=\) | \(2^5\) |
| \(=\) | \(32\) |
Condense with the product rule, then rewrite (base 10).
| \(\log\big(x(x-3)\big)\) | \(=\) | \(1\) |
| \(x(x-3)\) | \(=\) | \(10^1=10\) |
| \(x^2-3x-10\) | \(=\) | \(0\) |
| \((x-5)(x+2)\) | \(=\) | \(0\) |
Candidates \(x=5,\ -2\). Reject \(x=-2\) (makes \(\log x\) undefined), so \(x=5\).
Exponentiate with base \(e\).
| \(2x-1\) | \(=\) | \(e^3\) |
| \(x\) | \(=\) | \(\dfrac{e^3+1}{2}\approx 10.5\) |
Move logs together, condense, then rewrite.
| \(\log_3(x-8)+\log_3 x\) | \(=\) | \(2\) |
| \(x(x-8)\) | \(=\) | \(3^2=9\) |
| \(x^2-8x-9\) | \(=\) | \(0\) |
| \((x-9)(x+1)\) | \(=\) | \(0\) |
Candidates \(x=9,\ -1\); reject \(x=-1\), so \(x=9\).
Common pitfalls
Frequently asked questions
How do you solve a logarithmic equation?
Condense to one logarithm, rewrite in exponential form, solve, and check each solution against the domain.
What is an extraneous solution?
A value that satisfies the transformed equation but not the original — here, one that makes a log's argument zero or negative. Reject it.
Why do you have to check solutions?
Because logs require positive inputs. Solving can produce candidates that violate this, so each must be verified.
How do you rewrite a log equation without logs?
Use \(\log_b x=c\iff x=b^c\). Condense to a single log first, then exponentiate both sides with base \(b\).