Mathematics · Algebra

Laws of Logarithms

Q: What are the three laws of logarithms?

Product rule: $\log{}(x\cdot{}y) = \log{} x + \log{} y$, a product in the argument becomes a sum. Quotient rule: $\log{}(\frac{x}{y}) = \log{} x - \log{} y$, a fraction becomes a difference. Power rule: $\log{}(x^{r}) = r\cdot{}\log{} x$, the exponent moves in front of the logarithm as a factor. All three hold in every base (ln, lg, log₂) and for positive arguments. Number example with the base-10 logarithm: $lg(1000\cdot{}100) = lg 1000 + lg 100 = 3 + 2 = 5$, and indeed $10^{5} = 100000 = 1000\cdot{}100$ ✓. The laws are the translation of the laws of exponents into the logarithm world: because exponents add when powers are multiplied, logarithms add, since a logarithm IS an exponent.

The laws of logarithms turn products into sums, quotients into differences and powers into multiples.

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Formula

lo g_{b} (x \cdot y) = lo g_{b} x + lo g_{b} y, lo g_{b} (x^{r}) = r \cdot lo g_{b} x

Dimensionless (algebra) · arguments x, y > 0

Variables & units

Variables & units
Symbol	Meaning	Unit
$log_{\mathrm{b}}$	Logarithm with base b (b > 0, b ≠ 1)	dimensionless
x, y	Positive arguments	dimensionless
$r$	Exponent, moves in front of the logarithm as a factor	dimensionless

LaTeX

\log_{b}(x \cdot y) = \log_{b} x + \log_{b} y, \quad \log_{b}(x^{r}) = r \cdot \log_{b} x

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Derivation & background

John Napier published the first logarithm tables in 1614; they turned multiplications into additions and revolutionized computation in astronomy and navigation. The three laws: $\log{}(x\cdot{}y) = \log{} x + \log{} y$ , $\log{}(\frac{x}{y}) = \log{} x - \log{} y$ , $\log{}(x^{r}) = r\cdot{}\log{} x$ . Additionally the change of base $\log{}_{b} x =\frac{ \ln{} x}{\ln{}} b$ . Important: there is no law for log(x + y).

Exam blueprint

Validity range

Hold for positive arguments x, $y > 0$ and bases $b > 0$ , $b \ne{} 1$ , in every base (ln, lg, log₂). There is no law for log(x + y).

Derivation steps

Logarithms translate the laws of exponents: log_b x is the exponent for x.

1With $x = b^{m}$ and $y = b^{n}$ , x·y = b^(m+n) (law of exponents).
2Taking logarithms gives $\log{}(x\cdot{}y) = m + n = \log{} x + \log{} y$ ; quotient and power analogously.

Rearrangements

Quotient

\log_{b}\frac{x}{y} = \log_{b} x - \log_{b} y

Division becomes subtraction.

Change of base

\log_{b} x = \frac{\ln x}{\ln b}

Makes every base accessible with the calculator.

Solve exponential equation

a^{x} = c \;\Rightarrow\; x = \frac{\ln c}{\ln a}

Taking logarithms brings x down from the exponent.

Task variant

Simplify ln(8) + ln(2) − ln(4).

$\ln{}(\frac{8\cdot{}2}{4}) = \ln{} 4 \approx{} 1.386$ .
Alternatively: $3\cdot{}\ln{} 2 + \ln{} 2 - 2\cdot{}\ln{} 2 = 2\cdot{}\ln{} 2 = \ln{} 4$ .

After how many years does a stock double at 3% growth per year?

1.03ᵗ = 2, so $t =\frac{ \ln{} 2}{\ln{}} 1.03 \approx{}\frac{ 0.693}{0.0296} \approx{} 23.4$ years.

Common mistakes

Splitting log(x + y) into log x + log y.

The law holds only for products: $\log{}(x\cdot{}y) = \log{} x + \log{} y$ .

Confusing $(\log{} x)^{2}$ with $\log{}(x^{2})$ .

$\log{}(x^{2}) = 2\cdot{}\log{} x$ ; $(\log{} x)^{2}$ is the square of the logarithm.

Taking logarithms of negative numbers.

log is defined only for positive arguments.

Swapping numerator and denominator in the change of base.

$\log{}_{b} x =\frac{ \ln{} x}{\ln{}} b$ , the base goes below.

Exam context

Exponential equations (decay, growth), manipulations in calculus and stochastics.

Formula cluster

Exponential and logarithm

Laws of exponents, laws of logarithms and the exponential function form one system.

(ln x)' = 1/x

Derivative of the Natural Logarithm

(eˣ)' = eˣ

Derivative of the Exponential Function

Worked example

2ˣ = 10: $x = \log{}_{2} 10 =\frac{ \ln{} 10}{\ln{}} 2 \approx{}\frac{ 2.303}{0.693} \approx{} 3.32$ . Check: $2^{3.32} \approx{} 9.98 \approx{} 10$ ✓. And: $lg(1000\cdot{}100) = lg 1000 + lg 100 = 3 + 2 = 5$ .

Applications

Solving exponential equations (half-life, doubling time), pH and decibel scales, compound interest, complexity analysis in computer science

How do you calculate with Laws of Logarithms?

Here is how to work through a typical Laws of Logarithms (log(x·y) = log x + log y) task step by step:

1
Task
Simplify ln(8) + ln(2) − ln(4).
Solution path
1. $\ln{}(\frac{8\cdot{}2}{4}) = \ln{} 4 \approx{} 1.386$ .
2. Alternatively: $3\cdot{}\ln{} 2 + \ln{} 2 - 2\cdot{}\ln{} 2 = 2\cdot{}\ln{} 2 = \ln{} 4$ .
2
Task
After how many years does a stock double at 3% growth per year?
Solution path
1.03ᵗ = 2, so $t =\frac{ \ln{} 2}{\ln{}} 1.03 \approx{}\frac{ 0.693}{0.0296} \approx{} 23.4$ years.

Frequently asked questions about Laws of Logarithms

What are the three laws of logarithms?+

Product rule: $\log{}(x\cdot{}y) = \log{} x + \log{} y$ , a product in the argument becomes a sum. Quotient rule: $\log{}(\frac{x}{y}) = \log{} x - \log{} y$ , a fraction becomes a difference. Power rule: $\log{}(x^{r}) = r\cdot{}\log{} x$ , the exponent moves in front of the logarithm as a factor. All three hold in every base (ln, lg, log₂) and for positive arguments. Number example with the base-10 logarithm: $lg(1000\cdot{}100) = lg 1000 + lg 100 = 3 + 2 = 5$ , and indeed $10^{5} = 100000 = 1000\cdot{}100$ ✓. The laws are the translation of the laws of exponents into the logarithm world: because exponents add when powers are multiplied, logarithms add, since a logarithm IS an exponent.

How do I solve an exponential equation like

2^{x} = 10

Take logarithms on both sides, because the power rule brings the x down from the exponent: 2ˣ = 10 becomes $x\cdot{}\ln{} 2 = \ln{} 10$ , so $x =\frac{ \ln{} 10}{\ln{}} 2 \approx{}\frac{ 2.303}{0.693} \approx{} 3.32$ . Check: $2^{3.32} \approx{} 9.98 \approx{} 10$ ✓. Which base you choose does not matter, ln and lg give the same x; you can also write the result directly as log₂ 10. The same pattern solves all growth and decay tasks: 1.03ᵗ = 2 (doubling at 3% interest) gives $t =\frac{ \ln{} 2}{\ln{}} 1.03 \approx{} 23.4$ years. Important: first isolate the power, then take logarithms. For 5·2ˣ = 40, first divide by 5 (2ˣ = 8, x = 3) rather than pulling apart log(5·2ˣ) without respecting the rules.

What is the change-of-base formula and when do I need it?+

The change-of-base formula reads log_b x = ln x/ln b (or with lg instead of ln, the quotient is the same). You need it whenever your calculator does not offer the desired base: you enter log₂ 10 as $\frac{\ln{} 10}{\ln{}} 2 \approx{} 3.32$ . It can be derived in two lines: from $b^{y} = x$ , taking logarithms gives $y\cdot{}\ln{} b = \ln{} x$ , so $y =\frac{ \ln{} x}{\ln{}} b$ . The formula also shows that all logarithm systems differ only by a constant factor; that is why logarithm curves of different bases look like vertically stretched copies of each other. Most common error: swapping numerator and denominator. Mnemonic: the base goes at the bottom of the fraction, just as it sits at the bottom in the symbol log_b.

Why is log(x + y) not log x + log y?+

Because the product law only translates products into sums, not sums into sums. The logarithm answers the question about the exponent, and exponents add when powers are MULTIPLIED $(b^{m}\cdot{}b^{n} = b^{(m+n)})$ , not when they are added. A number test destroys the misconception immediately: $lg(10 + 10) = lg 20 \approx{} 1.301$ , but $lg 10 + lg 10 = 2$ . There simply is no general simplification law for log(x + y); such expressions stay as they are or require other techniques like factoring (lg(50 + 50) = lg 100 = 2 only because you add first). The same warning applies to log(x − y). In exams, wrongly splitting sums is one of the most frequent point losses in logarithm tasks.

What is the difference between ln, lg and log₂?+

They are logarithms to different bases. ln is the natural logarithm with base $e \approx{} 2.718$ ; it is the standard in calculus, because only it has the clean derivative $\frac{1}{x}$ . lg is the base-10 logarithm; it fits orders of magnitude and sits inside $pH (pH = -lg[H_{3}O^{+}])$ , decibels and earthquake magnitudes. log₂ is the base-2 logarithm of computer science: memory sizes, search trees, the number of halving steps in binary search. All three obey the same laws of logarithms and differ only by constant factors, convertible via change of base: $\log{}_{2} x =\frac{ \ln{} x}{\ln{}} 2 \approx{} 1.443\cdot{}\ln{} x$ . Beware of notation: a bare "log" means lg (engineering), ln (university mathematics) or log₂ (computer science) depending on the field; when in doubt state the base.

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Which formula describes Laws of Logarithms?

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How do you rearrange Laws of Logarithms for Quotient?

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Which common mistake happens with Laws of Logarithms?

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Subject references and further reading

This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.

[1]OpenStax, College Algebra 2e, §6.3: Logarithmic Functions

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Symbol

Meaning

Unit

log_{\mathrm{b}}

Logarithm with base b (b > 0, b ≠ 1)

dimensionless

x, y

Positive arguments

dimensionless

r

Exponent, moves in front of the logarithm as a factor

dimensionless

Derivation & background

Exam blueprint

Validity range

Hold for positive arguments x, $y > 0$ and bases $b > 0$ , $b \ne{} 1$ , in every base (ln, lg, log₂). There is no law for log(x + y).

Derivation steps

Logarithms translate the laws of exponents: log_b x is the exponent for x.

1With $x = b^{m}$ and $y = b^{n}$ , x·y = b^(m+n) (law of exponents).
2Taking logarithms gives $\log{}(x\cdot{}y) = m + n = \log{} x + \log{} y$ ; quotient and power analogously.

How do you calculate with Laws of Logarithms?

Here is how to work through a typical Laws of Logarithms (log(x·y) = log x + log y) task step by step:

Task

Simplify ln(8) + ln(2) − ln(4).

Solution path

$\ln{}(\frac{8\cdot{}2}{4}) = \ln{} 4 \approx{} 1.386$ .
Alternatively: $3\cdot{}\ln{} 2 + \ln{} 2 - 2\cdot{}\ln{} 2 = 2\cdot{}\ln{} 2 = \ln{} 4$ .

Task

After how many years does a stock double at 3% growth per year?

Solution path

1.03ᵗ = 2, so $t =\frac{ \ln{} 2}{\ln{}} 1.03 \approx{}\frac{ 0.693}{0.0296} \approx{} 23.4$ years.

Frequently asked questions about Laws of Logarithms

What are the three laws of logarithms?+

How do I solve an exponential equation like

2^{x} = 10

What is the change-of-base formula and when do I need it?+

Why is log(x + y) not log x + log y?+

What is the difference between ln, lg and log₂?+

Retain Laws of Logarithms for exams

Create a curated FSRS exam set for log(x·y) = log x + log y: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.

Curated formula set · LaTeX · FSRS spaced repetition · import from Essential