Logarithmic Equations and Inequalities — Mathematics Knowledge System

Overview

Logarithmic equations contain logarithms of expressions involving unknowns. Solving them uses log properties (product, quotient, power rules) and the inverse relationship: log_b(x) = y iff b^y = x. Key: always check for extraneous solutions (logarithms require positive arguments).

Intuition

log_b(x) = y is just another way of writing b^y = x. To solve log_2(x) = 5, convert: 2^5 = x = 32. For equations like log(x) + log(x-3) = 1 (base 10), use log product rule: log(x(x-3)) = 1, so x(x-3) = 10. Solve the resulting polynomial, then check that all arguments were positive.

Formal Definition

Definition

Log laws: log_b(mn) = log_b(m) + log_b(n); log_b(m/n) = log_b(m) - log_b(n); log_b(m^p) = p*log_b(m); log_b(b) = 1; log_b(1) = 0. Change of base: log_b(x) = ln(x)/ln(b). To solve: combine logs, convert to exponential form, solve, check domain.

\log_b(mn) = \log_b m + \log_b n

Product rule

\log_b\left(\frac{m}{n}\right) = \log_b m - \log_b n

Quotient rule

\log_b(m^p) = p\,\log_b m

Power rule

\log_b x = \frac{\ln x}{\ln b}

Change of base

Notation

Notation	Meaning
$\log_b(x)$	Logarithm base b of x: exponent to which b must be raised to get x
$\log(x)$	Common logarithm (base 10) in most algebra contexts

Theorems

Theorem 1: Logarithm-Exponential Inverse

log_b(b^x) = x for all x; b^{log_b(x)} = x for x > 0. These are inverses. Domain of log_b is (0, inf); range is R.

Worked Examples

1
Combine logs: log_3((x+2)(x-4)) = 3.
$\log_3((x+2)(x-4)) = 3$
2
Convert to exponential: (x+2)(x-4) = 27.
$x^2 - 2x - 8 = 27 \Rightarrow x^2-2x-35=0$
3
Factor: (x-7)(x+5) = 0, so x = 7 or x = -5.
$x = 7 \text{ or } x = -5$
4
Check: x=7: log_3(9)+log_3(3)=2+1=3. OK. x=-5: log_3(-3) undefined. Reject.
$x = 7$

✓ Answer

x = 7

Practice Problems

Mediumapplication

Solve 2^{x+1} = 5 for x.

Common Mistakes

Common Mistake

Forgetting to check for extraneous solutions after solving.

Log equations can produce extraneous solutions after squaring or polynomial steps. Always substitute back and verify that all logarithm arguments are positive.

Quiz

The equation log_2(x) = -3 has solution:

-81/88No solution

Historical Background

Logarithms were invented by John Napier (1614) and simultaneously by Joost Burgi (1620) as computational tools to simplify multiplication (via log product rule: log(ab) = log a + log b). Henry Briggs and Napier collaborated to create common logarithm tables (base 10). The slide rule, used until the 1970s, is a physical embodiment of logarithm addition.

1614
John Napier publishes Mirifici Logarithmorum Canonis Descriptio
John Napier
1624
Henry Briggs publishes Arithmetica Logarithmica with base-10 log tables
Henry Briggs

Summary

log_b(x) = y iff b^y = x. Domain: x > 0; range: all reals.
Product: log(mn)=log(m)+log(n). Quotient: log(m/n)=log(m)-log(n). Power: log(m^p)=p*log(m).
To solve: combine using log laws, convert to exponential form, solve, check for extraneous solutions.
Change of base: log_b(x) = ln(x)/ln(b).

References

BookLarson, R. Algebra and Trigonometry. 9th ed. Brooks/Cole, 2013.