Logarithm Rule for Addition- Simplify Expressions
What Is the Logarithm Rule for Addition?
The logarithm rule for addition states that when you're adding two logarithms with the same base, you can combine them into a single logarithm of the product of the arguments.
Here's the formula:
logb(x) + logb(y) = logb(x × y)
This is also called the product rule for logarithms. It works in both directions — you can expand a single log into two, or compress two logs into one.
Why This Rule Exists
Logarithms convert multiplication into addition. That's the whole point of them. John Napier invented logs in 1614 specifically so astronomers could avoid grinding through tedious multi-digit multiplication.
The addition rule is the direct result of that core property. When you add two logs, you're undoing two separate exponentiations, which means you're multiplying the original numbers together.
How to Apply the Rule
Combining Two Logs into One
If you have log(3) + log(7) with the same base, combine them:
log(3) + log(7) = log(3 × 7) = log(21)
That's it. Multiply the numbers inside, keep the same base.
Expanding One Log into Two
The rule works in reverse too. If you have log(50), you can split it:
log(50) = log(5 × 10) = log(5) + log(10)
You split a log by factoring its argument into two numbers that multiply to it.
Common Mistake to Avoid
This rule only applies when the logs have the same base. You cannot combine:
log2(x) + log3(y)
There's no simplification for this. The bases must match. If they don't, you're stuck unless you can convert one to the other using the change of base formula.
Full Table of Logarithm Rules
| Rule Name | Formula | Example |
|---|---|---|
| Product Rule | logb(x) + logb(y) = logb(xy) | log₂(3) + log₂(5) = log₂(15) |
| Quotient Rule | logb(x) - logb(y) = logb(x/y) | log₂(12) - log₂(3) = log₂(4) |
| Power Rule | logb(xn) = n × logb(x) | log₂(8³) = 3 × log₂(8) |
| Change of Base | logb(x) = logk(x) / logk(b) | log₂(10) = log(10)/log(2) |
| Zero Rule | logb(1) = 0 | log₂(1) = 0 |
Getting Started: Solving a Problem
Let's simplify: log₂(8) + log₂(4)
Step 1: Apply the product rule
log₂(8) + log₂(4) = log₂(8 × 4)
Step 2: Multiply
log₂(32)
Step 3: Evaluate if needed
Since 2⁵ = 32, log₂(32) = 5
Now try expanding: log₃(81)
Step 1: Factor 81
81 = 9 × 9
Step 2: Apply the product rule in reverse
log₃(81) = log₃(9) + log₃(9)
Step 3: Evaluate
log₃(9) = 2 (since 3² = 9)
So log₃(81) = 2 + 2 = 4 ✓
When to Use This Rule
- Simplifying expressions with multiple logs before solving equations
- Solving logarithmic equations where terms need condensing
- Converting complex products into single log expressions for easier differentiation or integration
- Expanding single logs to isolate specific terms in algebraic manipulation
Natural Log vs Common Log
The rule works identically for any base. Whether you're using log₁₀ (common log), ln (natural log), or any base b, the product rule stays the same:
ln(x) + ln(y) = ln(xy)
ln(2) + ln(5) = ln(10) ≈ 2.303
No difference in how you apply it. The only thing that changes is the base when you evaluate numerically.
The Bottom Line
The log addition rule is straightforward: same base, add the logs, multiply the arguments. That's the entire concept.
What trips people up isn't understanding the rule — it's knowing when to apply it. If you're simplifying an expression, look for opportunities to combine or split. If you're solving an equation, use the rule to condense terms so you can eliminate the log.
Master this rule and the quotient rule, and you'll handle 80% of log manipulation problems without breaking a sweat.