Logarithm Properties for Simplifying and Solving

Log Rules as Parallel Tools to Exponent Rules

Logarithm properties are “translation rules” between multiplication/division/powers and addition/subtraction/scaling. They let you rewrite complicated expressions into forms that are easier to simplify or solve. Throughout, assume a fixed base (often 10 or e) unless a base is shown explicitly.

Validity Conditions (Do Not Skip These)

Log rules are algebraic identities, but they only apply when the logarithms involved are defined (in the real-number setting):

• Argument must be positive: if you see log(M), you must have M > 0.
• Base restrictions (if shown): for log_b(M), require b > 0, b ≠ 1, and M > 0.

When you rewrite using log properties, you should mentally track these conditions. For example, log(x-3) implies x > 3, and log(2-x) implies x < 2. If both appear in the same expression, the domain is their intersection.

(1) Product, Quotient, and Power Rules

Product Rule

Rule: log(MN) = log(M) + log(N)

Condition: M > 0 and N > 0 (so that MN > 0 and each separate log is defined).

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Use: turns multiplication inside a log into addition outside.

Quotient Rule

Rule: log(M/N) = log(M) - log(N)

Condition: M > 0 and N > 0 (also ensures M/N > 0).

Use: turns division inside a log into subtraction outside.

Power Rule

Rule: log(M^p) = p·log(M)

Condition: In real logs, require M > 0. The exponent p can be any real number, but note a common subtlety: if you start with log(M^p) and p is not an integer, then M^p itself already forces M > 0 in the real setting. In practice for algebra problems, keep M > 0 as the safe condition.

Use: turns exponents into coefficients, which is especially useful for “linear-looking” combinations.

(2) Expanding and Condensing Expressions (Step-by-Step)

Expanding: Break One Log into Several

Goal: rewrite a single log of a product/quotient/power into a sum/difference with coefficients.

Example A (expand fully): Expand log( (3x^2√y) / (5z^3) ).

Step 1: Split quotient.

log( (3x^2√y) / (5z^3) ) = log(3x^2√y) - log(5z^3)

Step 2: Split products.

log(3x^2√y) = log(3) + log(x^2) + log(√y)

log(5z^3) = log(5) + log(z^3)

Step 3: Pull powers out as coefficients. Use √y = y^(1/2).

log(x^2) = 2 log(x)

log(√y) = log(y^(1/2)) = (1/2) log(y)

log(z^3) = 3 log(z)

Combine:

log( (3x^2√y) / (5z^3) ) = [log 3 + 2 log x + (1/2) log y] - [log 5 + 3 log z]

= log 3 - log 5 + 2 log x + (1/2) log y - 3 log z

Domain conditions: x > 0, y > 0, z > 0.

Condensing: Combine Several Logs into One

Goal: rewrite a sum/difference of logs (possibly with coefficients) into a single logarithm.

Example B (condense): Condense 2 log(x) - (1/3) log(y) + log(5) - log(z).

Step 1: Move coefficients into exponents (reverse power rule).

2 log(x) = log(x^2)

(1/3) log(y) = log(y^(1/3))

So the expression becomes:

log(x^2) - log(y^(1/3)) + log(5) - log(z)

Step 2: Combine additions with product rule.

log(x^2) + log(5) = log(5x^2)

Step 3: Combine subtractions with quotient rule.

log(5x^2) - log(y^(1/3)) = log( (5x^2) / (y^(1/3)) )

log( (5x^2) / (y^(1/3)) ) - log(z) = log( (5x^2) / (z·y^(1/3)) )

Final condensed form:

log( (5x^2) / (z·y^(1/3)) )

Domain conditions: x > 0, y > 0, z > 0.

(3) How Log Properties Create Linear-Looking Combinations

Many algebra and modeling tasks become easier when multiplication turns into addition and powers turn into coefficients. This is why log expressions often look “linear” after rewriting.

Addition of Logs Corresponds to Multiplication

If a > 0 and b > 0, then:

log(a) + log(b) = log(ab)

This means a sum of log terms can represent a single log of a product. In data contexts, this is the algebra behind converting multiplicative relationships into additive ones.

Subtraction of Logs Corresponds to Division

log(a) - log(b) = log(a/b)

So a difference of log terms can represent a ratio.

Coefficients in Front of Logs Correspond to Powers

k·log(a) = log(a^k)

This is especially useful when you see expressions like:

3 log(x) - 2 log(y) + log(7)

which condenses to:

log( 7·x^3 / y^2 )

Notice how the outside linear combination (coefficients and plus/minus) becomes a single inside expression built from products, quotients, and powers.

(4) Common Mistakes and How to Avoid Them

Mistake 1: Splitting a Sum Inside a Log

False: log(a + b) = log(a) + log(b)

This is not a property of logarithms. There is no simplification rule that breaks addition inside a log into separate logs.

Quick check with numbers: Let a = 1, b = 1 (base 10).

log(1+1) = log(2) ≈ 0.3010

log(1) + log(1) = 0 + 0 = 0

Not equal, so the “rule” is invalid.

Mistake 2: Canceling Logs Incorrectly

Be careful with expressions like log(x)/log(x). This equals 1 only when it is defined, meaning x > 0 and x ≠ 1 (since log(1)=0 would make the denominator zero).

Mistake 3: Dropping Absolute Value Ideas from Non-Log Algebra

In some algebra contexts you may be used to identities like √(x^2)=|x|. With logs, the key restriction is simpler but strict: the argument must be positive. For example, log(x^2) is defined for x ≠ 0 (since x^2 > 0 when x ≠ 0), but log(x) is defined only for x > 0. So rewriting log(x^2) as 2log(x) silently changes the domain unless you already know x > 0.

Safe habit: when applying log(M^p)=p log(M), confirm that the rewritten form keeps the intended domain (often the problem implicitly assumes variables represent positive quantities, but not always).

Mistake 4: Applying Rules Across Different Bases

The product/quotient/power rules assume the same log function throughout. You cannot combine log_2(x) + log_10(y) into a single log without first rewriting them to a common base.

(5) Practice Sets (Expand and Condense)

Set A: Expand (write as a sum/difference of logs)

1. Expand fully: log( 12x^3y^2 )
2. Expand fully: log( (a^5 b) / (c^2) )
3. Expand fully: log( (5m√n) / (2p^4) )
4. Expand fully: log( (x^2 (y+1)) / z ) (expand as much as valid; do not split log(y+1).)
5. Expand fully with rational exponent: log( (r^(-2) s^(3/2)) / t )
6. Expand: log( ( (3x^2y)^4 ) / (z^2) )

Set B: Condense (write as a single logarithm)

1. Condense: log(2) + log(x) + log(y)
2. Condense: log(a) - log(b) + log(c)
3. Condense with coefficients: 3 log(m) - 2 log(n)
4. Condense with rational coefficients: (1/2) log(x) + (3/4) log(y) - log(5)
5. Condense: log(7) + 2 log(p) - (1/3) log(q) - log(r)
6. Condense: log(x^2) - log(x) + log(4) (simplify inside the final log if possible, and state any domain restriction you used.)

Set C: Mixed and Nested Powers (both directions)

1. Expand: log( ( (x^3 y^(-1))^2 ) / (√z) )
2. Condense: 2 log( (a^2)/b ) - (1/2) log(c)
3. Expand: log( ( (2u v^2)^3 ) / ( (w^(-1))^2 ) )
4. Condense: log(x) + log(x-1) - 2 log(x+1) (include domain conditions from the original expression.)
5. Decide whether each statement is valid for real logs; if valid, rewrite using properties; if not, explain what blocks it:
   • log(x^2) = 2 log(x)
   • log(x+y) = log(x) + log(y)
   • log( (x-2)(x+2) ) = log(x-2) + log(x+2)

Answer-Checking Tips (without fully solving for you)

• After condensing, re-expand your result and see if you recover the original expression.
• Track domains: expanding can introduce stricter conditions (for example, turning log(x^2) into 2log(x) forces x > 0 instead of x ≠ 0).
• Never split a sum or difference inside a single log unless it is actually a product/quotient disguised (e.g., factor first: x^2-1=(x-1)(x+1) can then use the product rule).