Condense Logarithms Calculator

Combine logarithmic expressions into a single logarithm using product, quotient, and power rules with step-by-step solutions.

Select Operation & Enter Values

Result

Condensed Form
log(21)
Original Expressionlog(3) + log(7)
Rule AppliedProduct Rule
Numerical Value1.3222

Step-by-Step Solution

log(a) + log(b) = log(a · b)

Understanding Logarithm Condensation

Condensing logarithms is the process of combining multiple logarithmic expressions into a single logarithm. This is the reverse of expanding logarithms. It relies on three fundamental properties of logarithms: the product rule, quotient rule, and power rule. Mastering these rules is essential for simplifying complex logarithmic expressions in algebra, calculus, and applied mathematics.

When you condense a logarithmic expression, you are essentially using the properties of logarithms in reverse. Instead of breaking a single logarithm into multiple terms (expanding), you combine multiple terms into one (condensing).

Logarithm Rules for Condensing

Product Rule

The sum of logs equals the log of the product.

log(a) + log(b) = log(a · b)

Quotient Rule

The difference of logs equals the log of the quotient.

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

Power Rule

A coefficient multiplied by a log becomes a power inside the log.

n · log(a) = log(a^n)

Change of Base

Convert between logarithm bases using division.

log_b(a) = log(a) / log(b)

Log of 1

The logarithm of 1 is always zero in any base.

log_b(1) = 0

Log of the Base

The logarithm of the base equals one.

log_b(b) = 1

Step-by-Step Condensation Process

  1. Apply the Power Rule first: Move any coefficients in front of logarithms as exponents inside. For example, 3 log(x) becomes log(x³).
  2. Apply the Product Rule: Combine logarithms that are added together. For example, log(a) + log(b) becomes log(ab).
  3. Apply the Quotient Rule: Combine logarithms that are subtracted. For example, log(a) - log(b) becomes log(a/b).
  4. Simplify: Simplify the resulting expression inside the logarithm if possible.

Worked Examples

Example 1: Product Rule

Condense log(5) + log(4):

Using the product rule: log(5) + log(4) = log(5 × 4) = log(20)

Example 2: Quotient Rule

Condense log(100) - log(4):

Using the quotient rule: log(100) - log(4) = log(100/4) = log(25)

Example 3: Power Rule

Condense 2 log(5):

Using the power rule: 2 log(5) = log(5²) = log(25)

Example 4: Combined

Condense 2 log(3) + 3 log(2):

Step 1 (power rule): log(3²) + log(2³) = log(9) + log(8)

Step 2 (product rule): log(9 × 8) = log(72)

Common Mistakes to Avoid

  • Logarithms can only be condensed when they have the same base.
  • log(a) + log(b) is NOT log(a + b). The correct answer is log(ab).
  • log(a) - log(b) is NOT log(a - b). The correct answer is log(a/b).
  • The power rule only works with coefficients that multiply the entire log expression.
  • Always check that the arguments of the logarithm remain positive after condensing.

Applications

Condensing logarithms is used in solving logarithmic equations, simplifying expressions in calculus, analyzing exponential growth and decay models, working with pH calculations in chemistry, decibel calculations in acoustics, and Richter scale computations in seismology. It is also a core skill for standardized tests including the SAT, ACT, and AP Calculus exams.