Getting the Variable Out
To solve an equation where the variable is inside a log, you must "Exponentiate" both sides to cancel the log.
Worked Examples
Example 1: Basic Log Equation
Solve: \(\log_2(x + 3) = 5\)
- Convert to exponential form: \(x + 3 = 2^5\).
- \(x + 3 = 32\).
- \(x = 29\).
- Result: 29
Example 2: Using Natural Logs
Solve: \(e^{2x} = 20\)
- Take the natural log (\(\ln\)) of both sides.
- \(\ln(e^{2x}) = \ln(20)\).
- \(2x = \ln(20)\).
- \(x = \frac{\ln(20)}{2} \approx \frac{2.99}{2} \approx 1.49\).
- Result: 1.49
Example 3: Checking for Extraneous Solutions
Solve: \(\log(x) + \log(x - 3) = 1\)
- Condense: \(\log(x(x-3)) = 1\).
- Exponentiate (base 10): \(x^2 - 3x = 10^1\).
- \(x^2 - 3x - 10 = 0\).
- Factor: \((x - 5)(x + 2) = 0\).
- Roots: \(x = 5, x = -2\).
- Warning: You cannot take the log of a negative number. \(x = -2\) is extraneous.
- Result: \(x = 5\)
The Bridge to Quantum Mechanics
In Quantum Mechanics, the Schrödinger equation is often written in "log-form" to simplify certain types of potential wells. When we solve for the energy of a particle "tunneling" through a barrier, the final answer often looks like \(\ln(\text{Transmission}) = -2\kappa L\). Solving this for the Transmission probability requires exactly the skills you learned here. Being able to move between exponents and logs is how we calculate the actual chance of a quantum event occurring.