Logarithmic Equations & Inequalities
A rigorous, step-by-step guide — from prerequisites to variable-base problems
1. Prerequisites Recap
1.1 Definition of the Logarithm
The logarithm is defined as the inverse of exponentiation:
$$\log_a b = c \;\Longleftrightarrow\; a^{c} = b, \qquad a > 0,\; a \neq 1,\; b > 0$$
Read this as: "$\log_a b$ is the exponent you must put on the base $a$ to obtain $b$." The three restrictions ($a>0$, $a\neq 1$, $b>0$) are non-negotiable — they come from the properties of the exponential function $a^x$.
1.2 Fundamental Properties
All rules below assume positive arguments and a positive base $\neq 1$.
| Property | Formula | Condition |
|---|---|---|
| Product rule | $\log_a(xy) = \log_a x + \log_a y$ | $x>0,\; y>0$ |
| Quotient rule | $\log_a\!\left(\dfrac{x}{y}\right) = \log_a x - \log_a y$ | $x>0,\; y>0$ |
| Power rule | $\log_a(x^{k}) = k\,\log_a x$ | $x>0$ |
| Change of base | $\log_a x = \dfrac{\log_b x}{\log_b a}$ | $b>0,\;b\neq 1$ |
⚠ Caution: The log of a sum is NOT the sum of the logs: $\log_a(x+y) \neq \log_a x + \log_a y$. Similarly, $(\log_a x)^r \neq r\,\log_a x$ — the power rule applies only when the argument is raised to a power, not the entire logarithm.
1.3 Monotonicity
The function $y = \log_a x$ is:
- Strictly increasing when $a > 1$ (e.g., $\log_2$, $\ln$, $\log_{10}$).
- Strictly decreasing when $0 < a < 1$ (e.g., $\log_{1/2}$, $\log_{0.3}$).
This fact is the engine behind every logarithmic inequality.
Interactive visualization — available in the full lesson
2. Domain of Validity (ОДЗ)
Before any algebraic manipulation, you must determine the set of $x$-values for which every logarithm in the equation/inequality is defined. This is the single most important procedural step.
Rule: For every expression $\log_{a(x)}\!\bigl(f(x)\bigr)$ in the problem, require: