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Logarithmic Equation Solver

log_b(x)=c · Sum of Logs · Linear in Log · Natural ln · Same-base equality

Solve equations where the variable appears inside a logarithm. Full step-by-step solution, domain check, and extraneous solution detection.

Quick Examples

Equation Type

Equation form: log_b(x) = c → x = bc

log₁₀(x) = 3

Equation form: a·log_b(x) + c = d

Note: enter d as a decimal (e.g. for 2log(x) = log(25), enter d = 1.39794 for log₁₀ or d = log(25) ≈ 1.39794). Or use the quick example button above.

2 · log₁₀(x) + 0 = d

Equation form: log_b(ax+p) + log_b(cx+q) = log_b(k) or = N

First log argument: (a·x + p)

Second log argument: (c·x + q)

RHS: log_b(k) = N means k = bN

log₁₀(x + 3) + log₁₀(x − 1) = 5

Equation form: ln(a·x + b) = c → x = (ec − b) / a

ln(2x − 1) = 4

Equation form: log_b(Ax² + Bx + C) = N → Ax² + Bx + C = bN, solve quadratic

log₃(x² − 8x) = 2
Log Rules Reference Card
Product rule: log_b(x·y) = log_b(x) + log_b(y) Quotient rule: log_b(x/y) = log_b(x) − log_b(y) Power rule: log_b(x^n) = n · log_b(x) Change of base: log_b(x) = ln(x) / ln(b) = log(x) / log(b) Inverse (exp): b^(log_b(x)) = x Inverse (log): log_b(b^x) = x Special values: log_b(1) = 0, log_b(b) = 1 Domain: log_b(x) defined only for x > 0, b > 0, b ≠ 1

What Are Logarithmic Equations?

A logarithmic equation is an equation in which the unknown variable appears as the argument of a logarithm. Examples range from the simple log(x) = 3 to more complex forms like log₂(x+3) + log₂(x−1) = 5. Solving these equations requires systematically applying logarithm rules to isolate the log, then using the inverse relationship between logarithms and exponentiation to recover x.

Logarithmic equations appear in many branches of science and engineering, from chemistry (pH calculation) to seismology (Richter scale) to acoustics (sound intensity in decibels) to finance (compound interest and time value of money).

Types of Logarithmic Equations

TypeFormMethod
Basiclog_b(x) = cExponentiate: x = b^c
Linear in loga·log_b(x) + c = dIsolate log, then exponentiate
Sum of logslog(f(x)) + log(g(x)) = NProduct rule, then solve polynomial
Natural logln(ax+b) = cExponentiate with e: ax+b = e^c
Same-base equalitylog_b(quadratic) = NSet quadratic = b^N, solve, check domain

The Five Core Logarithm Rules

Mastering logarithm rules is essential for solving log equations:

  • Product rule: log_b(xy) = log_b(x) + log_b(y). Use this to combine a sum of logs into a single log.
  • Quotient rule: log_b(x/y) = log_b(x) − log_b(y). Use this to combine a difference of logs.
  • Power rule: log_b(x^n) = n·log_b(x). Use this to bring coefficients in front of logs inside as exponents.
  • Change-of-base formula: log_b(x) = ln(x)/ln(b). Converts any base to natural log for numerical calculation.
  • Inverse property: b^(log_b(x)) = x. The fundamental tool for removing the logarithm once isolated.

Domain Restrictions and Extraneous Solutions

The most critical aspect of solving logarithmic equations is the domain restriction: log_b(u) is only defined when u > 0. This requirement applies to every logarithm argument in the equation simultaneously. When solving a sum-of-logs equation that leads to a quadratic, both roots must be checked individually — a root is extraneous if substituting it back makes any argument zero or negative.

For example, solving log(x) + log(x−3) = 1 yields x = 5 and x = −2. Substituting x = −2: log(−2) is undefined, so x = −2 is discarded. Only x = 5 is a valid solution.

Natural Logarithm vs Common Logarithm

The natural logarithm (ln) uses Euler's number e ≈ 2.71828 as its base. It appears naturally in calculus, differential equations, and growth/decay models. The common logarithm (log₁₀, often written simply as "log") uses base 10 and is used in pH measurement, the Richter scale, and decibel calculations. Both satisfy the same algebraic rules; they differ only in their base and the constant relating them: log(x) = ln(x) / ln(10) ≈ ln(x) / 2.3026.

Real-World Applications

  • pH calculation: pH = −log[H⁺]. Solving for hydrogen ion concentration from a known pH requires exponentiating: [H⁺] = 10^(−pH).
  • Richter scale: Earthquake magnitude M = log(I/I₀). Finding intensity ratio from magnitude: I/I₀ = 10^M.
  • Decibels: Sound level L = 10·log(P/P₀). Solving for power ratio from decibel level is a direct logarithmic equation.
  • Finance: Compound interest A = P(1+r)^t. Solving for time t requires taking logarithms: t = log(A/P) / log(1+r).
  • Information theory: Shannon entropy H = −∑ p·log₂(p). Solving for probabilities from entropy values involves logarithmic equations.
  • Radioactive decay: N(t) = N₀·e^(−λt). Solving for time uses the natural logarithm: t = ln(N/N₀) / (−λ).

Worked Example: log₂(x+3) + log₂(x−1) = 5

Step 1 — Combine using product rule: log₂((x+3)(x−1)) = 5

Step 2 — Exponentiate both sides (base 2): (x+3)(x−1) = 2⁵ = 32

Step 3 — Expand: x² + 2x − 3 = 32 → x² + 2x − 35 = 0

Step 4 — Factor/quadratic formula: (x+7)(x−5) = 0 → x = −7 or x = 5

Step 5 — Domain check: x = −7: log₂(−7+3) = log₂(−4) — undefined. x = 5: log₂(8) and log₂(4) — both valid.

Answer: x = 5 (x = −7 is extraneous)

Frequently Asked Questions

What is a logarithmic equation?
A logarithmic equation is any equation where the variable appears inside a logarithm, such as log(x) = 3 or log₂(x+3) + log₂(x−1) = 5. Solving requires applying log rules to isolate or combine the logs, then exponentiating to remove them, followed by a domain check to discard extraneous solutions.
What are the main logarithm rules used to solve equations?
The five core rules are: (1) Product: log_b(xy) = log_b(x) + log_b(y); (2) Quotient: log_b(x/y) = log_b(x) − log_b(y); (3) Power: log_b(x^n) = n·log_b(x); (4) Change-of-base: log_b(x) = ln(x)/ln(b); (5) Inverse: b^(log_b(x)) = x. When solving sum-of-logs equations, the product rule is used to combine logs on one side before exponentiating.
What are extraneous solutions in logarithmic equations?
Extraneous solutions are values of x that satisfy the rearranged polynomial equation but fail the domain check for logarithms. Since log_b(u) requires u > 0, any solution that makes any log argument non-positive must be discarded. They appear most often in sum-of-logs equations that reduce to quadratics, where one root may be negative or make an argument zero.
How do you solve log_b(x) = c?
Apply the definition of logarithm directly: log_b(x) = c means x = b^c. For example, log₁₀(x) = 3 gives x = 10³ = 1000. log₂(x) = 5 gives x = 2⁵ = 32. ln(x) = 4 gives x = e⁴ ≈ 54.598. The solution is always positive (since b^c > 0), so no domain check is needed for the pure basic form.
How do you solve equations with a sum of logarithms?
Use the product rule to combine the sum of logs: log(A) + log(B) = log(AB). Then exponentiate both sides to get AB = b^(RHS). Expand and solve the resulting polynomial equation. Finally, check every root against the domain — each original log argument must evaluate to a positive number. Discard any root that fails this check.
What is the difference between log and ln?
log (common logarithm) uses base 10, while ln (natural logarithm) uses base e ≈ 2.71828. They satisfy the same algebraic rules and are related by log(x) = ln(x) / ln(10) ≈ ln(x) / 2.3026. In science, ln appears in exponential growth/decay (half-life, compound interest), while log₁₀ appears in pH, Richter scale, and decibels.
How do you solve a·log_b(x) + c = d?
Isolate the log in two steps: subtract c (giving a·log_b(x) = d−c), then divide by a (giving log_b(x) = (d−c)/a). Finally exponentiate: x = b^((d−c)/a). For 2·log(x) = log(25): rewrite RHS as log(25) ≈ 1.39794, so log(x) = 0.69897, x = 10^0.69897 = 5. Equivalently, use the power rule: log(x²) = log(25) → x² = 25 → x = 5 (x = −5 extraneous).

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