Quick Revision Notes — Solving Logarithms

60-Second Summary

The core takeaways in under a minute

Solving logarithmic equations requires collapsing terms using log laws (product, quotient, power) and then converting to exponential form ($b^x = a$). The **Golden Rule** is to check every solution: reject any root that makes the argument of a logarithm negative or zero, or makes a variable base non-positive or equal to 1. Logarithmic and exponential functions are inverses; their graphs reflect across the diagonal line $y = x$.

Step-by-Step Solving Algorithm

Follow these steps to solve any log equation

1
Condense terms: Use properties to merge multiple logs into one: $\log_b(\text{expression}) = c$.
2
Convert to Exponent: Rewrite the equation as: $\text{expression} = b^c$.
3
Solve Algebra: Solve the linear or quadratic equation for $x$.
4
Domain Verification: Substitute roots back to verify: arguments $>0$, base $>0$, and base $\neq 1$.

Common Mistakes to Avoid

Watch out for these classic exam traps!

1
Forgetting to Reject Extraneous Roots

Just because $x = -2$ solves the algebraic quadratic equation doesn't make it a valid logarithm solution. Always plug it back in; if you get a term like $\log_5(-2)$, reject it immediately.

2
Variable base constraints

In equations like $\log_x(10) = 2$, base $x$ must be strictly positive and not 1. If algebra yields $x = -2$, reject it since the base cannot be negative.

3
Quadratic Substitution Errors

Confusing $(\log_a x)^2$ with $\log_a(x^2)$. Note that $(\log_a x)^2$ is the square of the entire logarithm and cannot be simplified using the power rule, whereas $\log_a(x^2) = 2\log_a x$.