Logarithm Properties Formula

Logarithm properties are the three fundamental rules of logarithms: the product rule _b(xy) = _b x + _b y, the quotient rule _b\!(x/y) = _b x.

The Formula

log⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_b x + \log_b y
log⁡b ⁣(xy)=log⁡bx−log⁡by\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y
log⁡b(xn)=nlog⁡bx\log_b(x^n) = n\log_b x

When to use: Logarithms were invented to turn hard operations into easy ones. Multiplication becomes addition, division becomes subtraction, and exponentiation becomes multiplication. This is why slide rules worked—they added lengths (logarithms) to multiply numbers.

Quick Example

log⁡2(8⋅4)=log⁡28+log⁡24=3+2=5\log_2(8 \cdot 4) = \log_2 8 + \log_2 4 = 3 + 2 = 5 Check: 8⋅4=32=258 \cdot 4 = 32 = 2^5. \checkmark
log⁡(x3)=3log⁡x\log(x^3) = 3\log x

Notation

log⁡b\log_b denotes logarithm base bb. When no base is written, log⁡\log typically means log⁡10\log_{10} (common log) or log⁡e\log_e (natural log) depending on context.

What This Formula Means

The three fundamental rules of logarithms: the product rule log⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_b x + \log_b y, the quotient rule log⁡b ⁣(xy)=log⁡bx−log⁡by\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y, and the power rule log⁡b(xn)=nlog⁡bx\log_b(x^n) = n\log_b x.

Logarithms were invented to turn hard operations into easy ones. Multiplication becomes addition, division becomes subtraction, and exponentiation becomes multiplication. This is why slide rules worked—they added lengths (logarithms) to multiply numbers.

Formal View

log⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_b x + \log_b y; log⁡b ⁣(xy)=log⁡bx−log⁡by\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y; log⁡b(xn)=nlog⁡bx\log_b(x^n) = n\log_b x; all follow from ba+c=ba⋅bcb^{a+c} = b^a \cdot b^c

Worked Examples

Example 1

easy
Expand log⁡2(8x3)\log_2(8x^3) using logarithm properties.

Answer

3+3log⁡2x3 + 3\log_2 x

First step

1
Apply the product rule: log⁡2(8x3)=log⁡28+log⁡2x3\log_2(8x^3) = \log_2 8 + \log_2 x^3.

Full solution

  1. 2
    Evaluate log⁡28=3\log_2 8 = 3 and apply the power rule: log⁡2x3=3log⁡2x\log_2 x^3 = 3\log_2 x.
  2. 3
    Result: 3+3log⁡2x3 + 3\log_2 x.
The product rule log⁡b(MN)=log⁡bM+log⁡bN\log_b(MN) = \log_b M + \log_b N and power rule log⁡b(Mp)=plog⁡bM\log_b(M^p) = p\log_b M are the two most commonly used expansion properties.

Example 2

medium
Condense 2ln⁡x−12ln⁡(x+1)+3ln⁡y2\ln x - \frac{1}{2}\ln(x + 1) + 3\ln y into a single logarithm.

Example 3

medium
Use the change of base formula to evaluate log⁡320\log_3 20 to three decimal places.

Common Mistakes

  • Turning log⁡(x+y)\log(x+y) into log⁡x+log⁡y\log x+\log y - the sum rule applies to products log⁡(xy)\log(xy), never sums.
  • Leaving the exponent in place - the power rule pulls nn out front as nlog⁡bxn\log_b x.
  • Reversing quotient and product - division gives subtraction, multiplication gives addition.

Why This Formula Matters

These three rules are the only legal way to move a variable out of an exponent, so they underpin solving every exponential equation and modeling growth/decay and pH/decibel scales. Inventing a 'log of a sum' rule is the single most common precalculus error and silently corrupts the whole solution. Recognizing it by "Is the log's argument a product, quotient, or power I can split into add, subtract, or a front multiplier?" — rather than by familiar numbers — is what lets a student tell it apart from exponent rules and change of base and invented log-of-a-sum rule in a mixed problem set.

Frequently Asked Questions

What is the Logarithm Properties formula?

The three fundamental rules of logarithms: the product rule log⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_b x + \log_b y, the quotient rule log⁡b ⁣(xy)=log⁡bx−log⁡by\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y, and the power rule log⁡b(xn)=nlog⁡bx\log_b(x^n) = n\log_b x.

How do you use the Logarithm Properties formula?

Logarithms were invented to turn hard operations into easy ones. Multiplication becomes addition, division becomes subtraction, and exponentiation becomes multiplication. This is why slide rules worked—they added lengths (logarithms) to multiply numbers.

What do the symbols mean in the Logarithm Properties formula?

log⁡b\log_b denotes logarithm base bb. When no base is written, log⁡\log typically means log⁡10\log_{10} (common log) or log⁡e\log_e (natural log) depending on context.

Why is the Logarithm Properties formula important in Math?

These three rules are the only legal way to move a variable out of an exponent, so they underpin solving every exponential equation and modeling growth/decay and pH/decibel scales. Inventing a 'log of a sum' rule is the single most common precalculus error and silently corrupts the whole solution. Recognizing it by "Is the log's argument a product, quotient, or power I can split into add, subtract, or a front multiplier?" — rather than by familiar numbers — is what lets a student tell it apart from exponent rules and change of base and invented log-of-a-sum rule in a mixed problem set.

What do students get wrong about Logarithm Properties?

The procedure for logarithm properties is the easy part; the trap is turning log⁡(x+y)\log(x+y) into log⁡x+log⁡y\log x+\log y. Asking "Is the log's argument a product, quotient, or power I can split into add, subtract, or a front multiplier?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Logarithm Properties formula?

Before studying the Logarithm Properties formula, you should understand: logarithm.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Exponents and Logarithms: Rules, Proofs, and Applications →
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