Logarithm Laws Calculator
Apply logarithm laws: product, quotient, power, change of base, and natural log rules.
Select Law
Presets
Key Identities
logᵦ(1) = 0
b⁰ = 1 for any b
logᵦ(b) = 1
b¹ = b for any b
b^logᵦ(x) = x
exponential cancellation
logᵦ(bⁿ) = n
b^n inside base-b log
What Is the Logarithm Laws Calculator?
This calculator applies all six fundamental logarithm operations, product, quotient, power, change of base, expand, and condense, with numerical verification and step-by-step working. Enter real values and see both sides of the identity evaluated to confirm equality.
- ›Six law modes, product, quotient, power, change of base, expand complex expressions, and condense into a single log.
- ›Numerical verification, both the left-hand side and right-hand side are computed numerically so you can confirm they are equal.
- ›Step-by-step working, collapsible panel shows how each intermediate value is calculated.
- ›Key identities reference panel, always-visible reminder of logᵦ(1)=0, logᵦ(b)=1, and the cancellation identities.
- ›Preset examples, five pre-loaded problems covering common exam question patterns.
- ›localStorage persistence, your last inputs are restored on return.
Formula
Product Rule
logᵦ(M · N) = logᵦ(M) + logᵦ(N)
Quotient Rule
logᵦ(M / N) = logᵦ(M) − logᵦ(N)
Power Rule
logᵦ(Mⁿ) = n · logᵦ(M)
Change of Base
logᵦ(x) = logₐ(x) / logₐ(b) (any valid base a)
Expand Expression
logᵦ(xᵃ · yᵇ / zᶜ) = a·logᵦ(x) + b·logᵦ(y) − c·logᵦ(z)
| Symbol | Name | Description |
|---|---|---|
| M, N | Arguments | Positive real numbers whose logarithms are being combined |
| n | Exponent | Any real number, becomes a coefficient via the power rule |
| b | Base | The logarithm base (b > 0, b ≠ 1) |
| a | New base | The target base for change-of-base conversion |
| a,b,c | Exponents | Powers of x, y, z in the expand/condense expression |
Special Identities
logᵦ(1) = 0 (b⁰ = 1 for any valid b)
logᵦ(b) = 1 (b¹ = b for any valid b)
b^logᵦ(x) = x (exponential cancellation)
logᵦ(bⁿ) = n (power inside base-b log)
How to Use
- 1Select a law: Click one of the six law tabs: Product, Quotient, Power, Change of Base, Expand Expression, or Condense Logs.
- 2Select the base b: Choose base 10, base e (natural log), base 2, base 3, or base 5. For Change of Base, also select the target base a.
- 3Enter values: Enter M and N for product/quotient, M and exponent n for power, or x,y,z and exponents for expand. See the on-screen labels.
- 4Press Enter or click Apply Law: Results appear showing both the left- and right-hand sides evaluated numerically to 6 decimal places.
- 5Verify equality: The result card shows LHS = RHS with a checkmark (or approximate symbol) confirming the law holds for your inputs.
- 6Expand Step-by-step: Click the collapsible panel to see every intermediate calculation, including individual log values and how they combine.
Example Calculation
Product Rule, log₃(81 × 27) = log₃(81) + log₃(27)
Given: base b = 3, M = 81, N = 27
Step 1: Apply product rule
log₃(81 × 27) = log₃(81) + log₃(27)
Step 2: Evaluate each term
log₃(81) = log₃(3⁴) = 4
log₃(27) = log₃(3³) = 3
Step 3: Sum
4 + 3 = 7
Step 4: Direct verification
81 × 27 = 2187 = 3⁷
log₃(2187) = 7 ✓ Both sides equal 7
| Law | Expression | Simplifies To | Value |
|---|---|---|---|
| Product ★ | log₃(81 × 27) | log₃(81) + log₃(27) | 4 + 3 = 7 |
| Quotient | log₂(8 / 4) | log₂(8) − log₂(4) | 3 − 2 = 1 |
| Power | log₁₀(1000²) | 2 · log₁₀(1000) | 2 × 3 = 6 |
| Change base | log₅(125) | ln(125) / ln(5) | 3.000 |
| Condense | 2·log(x) + 3·log(y) | log(x² · y³) | single log |
Why both sides must be equal
Each logarithm law is an algebraic identity, it is always true for any valid inputs. The calculator evaluates both sides numerically as a double-check. If LHS ≠ RHS, there is an input error (such as a non-positive argument).
Understanding Logarithm Laws
Why Logarithm Laws Exist
Logarithm laws are consequences of exponent laws. Because logᵦ(x) = y means b^y = x, every exponent rule has a corresponding logarithm rule:
- ›Exponent product rule b^(m+n) = b^m · b^n → Log product rule: logᵦ(M·N) = logᵦ(M) + logᵦ(N)
- ›Exponent quotient rule b^(m−n) = b^m / b^n → Log quotient rule: logᵦ(M/N) = logᵦ(M) − logᵦ(N)
- ›Exponent power rule (b^m)^n = b^(mn) → Log power rule: logᵦ(M^n) = n · logᵦ(M)
Understanding this connection makes the laws intuitive: you are not memorising arbitrary rules but recognising a translation between the language of exponents and the language of logarithms.
Expanding vs Condensing, Two Directions
Logarithm laws are used in two directions depending on the goal:
- ›Expanding, split a complex log into simpler parts. Used in integration (logarithmic differentiation) and proving identities. Example: log(x²y³/z) = 2log(x) + 3log(y) − log(z).
- ›Condensing, combine separate logs into a single log. Used when solving equations like 2log(x) + log(3) = log(12). Example: log(x²) + log(3) = log(3x²).
Solving Exponential Equations with Logarithm Laws
Logarithm laws are the primary tool for solving equations where the unknown is in an exponent. The strategy is always: take the logarithm of both sides, then use the power rule to bring the exponent down as a coefficient.
Solve: 5^x = 200
Take ln of both sides: ln(5^x) = ln(200)
Power rule: x · ln(5) = ln(200)
Divide: x = ln(200) / ln(5) = 5.2983 / 1.6094 = 3.292
The product rule as multiplication-to-addition conversion
Before electronic calculators, the product rule was the main practical use of logarithms. To multiply 5,481 × 7,296, look up log(5481) and log(7296), add them, then look up the anti-log. This converts a difficult multiplication into two table lookups and one addition. Slide rules mechanically implemented this using a logarithmic physical scale.
Logarithm Laws in Calculus
In calculus, logarithm laws appear in three major contexts:
- ›Logarithmic differentiation, to differentiate complicated products/quotients/powers, take ln of both sides first, apply log laws to expand, then differentiate implicitly.
- ›Integration by parts and substitution, ∫ ln(x) dx = x·ln(x) − x + C uses the structure of logarithms.
- ›L'Hôpital's rule and limits, lim x→0⁺ x·ln(x) = 0 uses the power rule implicitly to simplify the limit.
Frequently Asked Questions
What are the three main logarithm laws?
- ›Product rule: logᵦ(M·N) = logᵦ(M) + logᵦ(N). Multiplication inside becomes addition outside.
- ›Quotient rule: logᵦ(M/N) = logᵦ(M) − logᵦ(N). Division inside becomes subtraction outside.
- ›Power rule: logᵦ(Mⁿ) = n · logᵦ(M). Exponents become coefficients.
A fourth law, the change-of-base formula, lets you evaluate any base using ln or log₁₀.
When do I use the product rule vs quotient rule?
Use the product rule when the argument is a product (multiplication): logᵦ(AB) = logᵦ(A) + logᵦ(B).
Use the quotient rule when the argument is a fraction (division): logᵦ(A/B) = logᵦ(A) − logᵦ(B).
For combined expressions like logᵦ(x²y/z³), apply the product rule to the numerator and quotient rule for the denominator, then the power rule for exponents:
What is the change-of-base formula and why is it needed?
The change-of-base formula is: logᵦ(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b).
It is needed because calculators and programming languages only provide natural log (ln) and common log (log₁₀). To compute log₇(300), for example:
Verification: 7^2.931 ≈ 300. The formula works because both ln and log₁₀ are themselves related by a constant factor.
How do I expand a logarithm of a fraction with exponents?
Apply the laws in order: quotient rule first, then product rule, then power rule:
- ›Step 1 (quotient): logᵦ(x³y² / z⁴) = logᵦ(x³y²) − logᵦ(z⁴)
- ›Step 2 (product): = logᵦ(x³) + logᵦ(y²) − logᵦ(z⁴)
- ›Step 3 (power): = 3·logᵦ(x) + 2·logᵦ(y) − 4·logᵦ(z)
Use the Expand mode in this calculator to see this applied with actual numbers.
How do I condense multiple logarithms into one?
Work in the reverse direction, convert coefficients back to exponents (power rule in reverse), then combine using product/quotient rules:
2·log(x) + 3·log(y) − log(z)
= log(x²) + log(y³) − log(z) (power rule)
= log(x²y³) − log(z) (product rule)
= log(x²y³/z) (quotient rule)
Why does the power rule let you move exponents outside the log?
The power rule follows directly from the definition of logarithms and the exponent product rule.
If logᵦ(M) = t, then b^t = M. So M^n = (b^t)^n = b^(tn). Therefore logᵦ(M^n) = tn = n · logᵦ(M).
This is why solving exponential equations like 5^x = 200 uses the power rule: take log of both sides, then x · log(5) = log(200), so x = log(200)/log(5).
What common mistakes do students make with log laws?
- ›log(M + N) ≠ log(M) + log(N), the product rule applies to multiplication, not addition inside the log.
- ›log(M) / log(N) ≠ log(M/N), log(M/N) = log(M) − log(N), not a division of logs.
- ›[log(M)]^n ≠ n·log(M), the power rule applies to logᵦ(M^n), where n is the exponent of the argument, not an exponent on the log itself.
- ›Forgetting the base must be positive and not equal to 1.
Does the calculator save my inputs?
Yes, inputs are saved to your browser's localStorage after each calculation. Data is restored when you return to the page. Click Reset to clear all fields and remove saved data. No data is sent to any server.