Properties of Logarithms

11 min read

Quick Reference

Try This First

Before any rule appears, work this with pencil and paper.

A logarithm is an exponent. The statement $\log_2 8 = 3$ means $2^3 = 8$: the logarithm answers “what exponent on the base produces this number”. Use that idea to fill the table. For each row, find the exponent on $2$ that produces the given number, then add the two exponents in the last row.

Predict: before you compute the third row, write down what you expect $\log_2 32$ to equal. Will it relate to the first two answers, and how?

The Idea Everything Comes From

A logarithm reverses an exponential. For a base $b > 0$ with $b \neq 1$,

$$\log_b M = x \quad \text{means exactly} \quad b^x = M.$$

In words, $\log_b M$ is the exponent that the base $b$ needs in order to produce $M$. Read as a process, the logarithm function takes an input $M$ and gives back the exponent that builds it.

Because logarithms are exponents, every logarithm rule is an exponent rule wearing different clothing. To translate, write each quantity as a power of the same base. If $M = b^x$ and $N = b^y$, then $x = \log_b M$ and $y = \log_b N$. Whatever the exponent laws do to $x$ and $y$, the logarithm rules do to $\log_b M$ and $\log_b N$. A student who holds onto this translation can rebuild any logarithm rule from the matching exponent rule, with no separate memorizing. There is no need to rush this page: the time spent making the translation concrete pays back on every rule below.

The practical payoff is large. Logarithms turn multiplication into addition and powers into multiplication. Before calculators, this is how engineers and scientists carried out long computations using slide rules and printed log tables. The same transformations reappear in calculus (the derivative of $\ln x$), in statistics (log-likelihoods), and in every formula for exponential growth or decay.

Prerequisite Hub

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        A["Laws of<br/>Exponents"]
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    subgraph ThisSkill["This Skill"]
        B["Properties of<br/>Logarithms"]
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    subgraph Unlocks
        C["Exponential and<br/>Logarithmic Equations"]
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    A --> B
    B --> C

    style B fill:#d1fae5,stroke:#a565f0,stroke-width:3px

    click A "exponent-laws.html"
    click B "logarithm-rules.html"

Builds on

Unlocks

The scope for this node lists no downstream skills yet. In the course sequence, the natural next step is solving exponential and logarithmic equations, where condensing with these rules is the first move. That node will be linked here once it is authored.

There are no cross-course prerequisites for this node.

Prerequisite Self-Check

Official Definitions

The product rule and the power rule below are stated exactly as in the primary source.

The Product Rule for Logarithms. The logarithm of a product is the sum of the logarithms: $\log_b(MN) = \log_b(M) + \log_b(N)$.

Source: OpenStax Precalculus 2e, Section 4.5 Logarithmic Properties.

The Power Rule for Logarithms. The logarithm of a power is the exponent times the logarithm of the base: $\log_b(M^n) = n \log_b(M)$.

Source: OpenStax Precalculus 2e, Section 4.5 Logarithmic Properties.

Section 4.5 also presents the quotient rule, which divides inside and subtracts outside. It is derived below from the same exponent reasoning, so the reader sees that it is forced by the definition rather than a separate fact to memorize.

Key Theorem

Change-of-Base Formula. For any positive real numbers $M$, $b$, and $n$ (with $b \neq 1$ and $n \neq 1$), $$\log_b(M) = \frac{\log_n(M)}{\log_n(b)},$$ allowing a logarithm in any base to be rewritten using common or natural logarithms.

Source: OpenStax Precalculus 2e, Section 4.5 Logarithmic Properties.

In practice $n$ is chosen to be $10$ or $e$, because a calculator has a common-log button ($\log$, base $10$) and a natural-log button ($\ln$, base $e$). With that choice, $$\log_b(M) = \frac{\log M}{\log b} = \frac{\ln M}{\ln b}.$$

Quick Reference

Let $M, N > 0$, let $b > 0$ with $b \neq 1$, and let $p$ be any real number.

Worked Examples

Product rule: multiplication becomes addition

When two positive quantities multiply inside a logarithm, their separate exponents add.

$$\log_b(MN) = \log_b(b^x \cdot b^y) = \log_b(b^{x+y}) = x + y = \log_b M + \log_b N$$

Example 1. Expand $\log_3(27x)$.

Predict: $27$ is a power of $3$, so its logarithm is a whole number, and the rule should split the product into a sum.

$$\log_3(27x) = \log_3 27 + \log_3 x = 3 + \log_3 x \qquad (\text{since } 3^3 = 27)$$

Example 2. Condense $\log 5 + \log 8$ into a single logarithm.

$$\log 5 + \log 8 = \log(5 \cdot 8) = \log 40$$

Quotient rule: division becomes subtraction

Dividing inside a logarithm subtracts the exponents outside. The derivation reuses the quotient rule for exponents.

$$\log_b\!\left(\frac{M}{N}\right) = \log_b\!\left(\frac{b^x}{b^y}\right) = \log_b(b^{x-y}) = x - y = \log_b M - \log_b N$$

Example 3. Expand $\ln\!\left(\dfrac{e^3}{x}\right)$.

Predict: the numerator $e^3$ has a clean natural logarithm, so expect a whole number minus a logarithm.

$$\ln\!\left(\frac{e^3}{x}\right) = \ln e^3 - \ln x = 3 - \ln x$$

Example 4. Condense $\ln 12 - \ln 4$.

$$\ln 12 - \ln 4 = \ln\!\left(\frac{12}{4}\right) = \ln 3$$

Power rule: an exponent moves out front

A power inside a logarithm becomes a coefficient outside, because raising to a power multiplies the exponent.

$$\log_b(M^p) = \log_b((b^x)^p) = \log_b(b^{xp}) = xp = p \log_b M$$

Example 5. Expand $\log_2(x^5)$.

Predict: the exponent $5$ should come out front as a coefficient.

$$\log_2(x^5) = 5 \log_2 x$$

Example 6. Rewrite $\tfrac{1}{2}\ln x$ as a single logarithm.

Read the coefficient $\tfrac{1}{2}$ as an exponent: a one-half power is a square root.

$$\frac{1}{2}\ln x = \ln x^{1/2} = \ln \sqrt{x}$$

Example 7. Expand $\log \sqrt[3]{x^2}$.

First rewrite the radical as a fractional exponent, then apply the power rule.

$$\log \sqrt[3]{x^2} = \log x^{2/3} = \frac{2}{3}\log x$$

Change of base: evaluating a logarithm in any base

A calculator computes only base $10$ and base $e$. The change-of-base formula routes every other base through one of those.

Example 8. Evaluate $\log_5 50$ to three decimal places.

Predict: $5^2 = 25$ and $5^3 = 125$, and $50$ sits between them but closer to $25$, so the answer should be a little above $2$.

$$\log_5 50 = \frac{\ln 50}{\ln 5} \approx \frac{3.912}{1.609} \approx 2.431$$

Check: $5^{2.431} \approx 50$, and $2.431$ falls between $2$ and $3$ as predicted. The prediction matches.

Combining the rules in one direction: expanding

Expanding writes one complicated logarithm as a sum and difference of simpler logarithms. It uses every rule at once.

Example 9. Expand $\ln\!\left(\dfrac{x^3 \sqrt{y}}{z^4}\right)$.

Step 1. Quotient rule separates the top from the bottom:

$$\ln(x^3 \sqrt{y}) - \ln(z^4)$$

Step 2. Product rule separates the two factors on top:

$$\ln(x^3) + \ln(\sqrt{y}) - \ln(z^4)$$

Step 3. Power rule moves each exponent to the front (and $\sqrt{y} = y^{1/2}$):

$$3\ln x + \frac{1}{2}\ln y - 4\ln z$$

Combining the rules in the other direction: condensing

Condensing reverses the process, collapsing a sum and difference of logarithms into one logarithm. The power rule is applied first so that every term is a plain logarithm before combining.

Example 10. Condense $2\log_3 x - \log_3(x+1) + \tfrac{1}{3}\log_3(x+2)$.

Step 1. Power rule moves each coefficient back into an exponent:

$$\log_3(x^2) - \log_3(x+1) + \log_3\big((x+2)^{1/3}\big)$$

Step 2. Product and quotient rules combine the three logarithms into one (added terms go in the numerator, the subtracted term goes in the denominator):

$$\log_3\!\left(\frac{x^2 (x+2)^{1/3}}{x+1}\right)$$

Common Misconceptions

Predict-then-check drill

For each expression, predict the correct simplified form, then check by substituting a number.

Leveled Practice

Mastery Checklist

Novice (Level 1-2):

• I can read $\log_b M = x$ as the statement $b^x = M$ and evaluate simple logarithms.
• I can apply the product rule to expand $\log_b(MN)$ into a sum.
• I can apply the quotient rule to expand $\log_b(M/N)$ into a difference.
• I can apply the power rule to move an exponent out front.

Competent (Level 3-4):

• I can use the change-of-base formula to evaluate a logarithm in any base with a calculator.
• I can fully expand a complex logarithm using all three rules in the right order.
• I can condense a sum and difference of logarithms into a single logarithm.

Proficient (Level 5):

• I can derive each logarithm rule from the matching exponent rule rather than reciting it.
• I can justify identities such as $\log_b a = \dfrac{1}{\log_a b}$ from the change-of-base formula.

Mental Model

A logarithm is an exponent counter. The question $\log_b M$ asks: how many factors of $b$ multiply together to make $M$.

From that single view, the rules read off directly:

• To build $MN$, gather the $b$-factors that make $M$ and the $b$-factors that make $N$. The counts pour together, so $\log_b(MN) = \log_b M + \log_b N$.
• To build $\dfrac{M}{N}$, remove from the $M$-factors the ones that match the $N$-factors. The counts subtract, so $\log_b\!\left(\dfrac{M}{N}\right) = \log_b M - \log_b N$.
• To build $M^p$, make $p$ copies of the $M$-factors. The count multiplies, so $\log_b(M^p) = p \log_b M$.

This is why logarithms turned multiplication into addition for slide-rule users. Multiplying two numbers was replaced by reading each number’s count from a table, adding the counts, and reading the result back. The rules on this page are what made that process possible.

Connections

Looking back:

• Laws of Exponents supply every rule on this page in their original exponent form. The product rule for logarithms is $b^x \cdot b^y = b^{x+y}$ read backward.

Looking ahead:

• Solving exponential and logarithmic equations begins by condensing with these rules and then converting between logarithm form and exponential form.
• Exponential growth and decay models such as $A = A_0 e^{kt}$ are solved for the time $t$ by taking $\ln$ of both sides and applying the power rule.

Toward calculus (MATH161):

• The derivative of the natural logarithm is $\dfrac{d}{dx}\ln x = \dfrac{1}{x}$, and the log rules appear inside the chain rule as $\dfrac{d}{dx}\ln(f(x)) = \dfrac{f'(x)}{f(x)}$.
• Logarithmic differentiation takes $\ln$ of both sides of a product or quotient of many factors, applies these rules to split it into a sum, and only then differentiates. The technique uses every rule on this page at once.

Real-world connections:

• Log scales compress huge ranges into readable numbers: decibels in acoustics, the Richter scale in seismology, pH in chemistry, star magnitude in astronomy, and entropy in information theory. Each one is a logarithm, and the rules here are how those scales are combined and compared.

Resources

• Primary text: OpenStax Precalculus 2e, Section 4.5 Logarithmic Properties. https://openstax.org/books/precalculus-2e/pages/4-5-logarithmic-properties
• Background on logarithmic functions: OpenStax Precalculus 2e, Section 4.3 Logarithmic Functions. https://openstax.org/books/precalculus-2e/pages/4-3-logarithmic-functions

Back to Precalculus Skills | Previous: Laws of Exponents

Last updated: 2026-06-16