Logarithms — Definition, Rules, Examples

The Question "What Power Gives This Number?"

Multiplication makes numbers bigger fast. Exponentiation makes them bigger faster. Logarithms are the inverse — they tell you what power produced a number, taking a huge value and returning the small exponent that built it.

Before pocket calculators, logarithms let astronomers replace huge multiplications with additions of small numbers (a 24-hour calculation turned into a 30-minute one). Today they show up in earthquake magnitudes, pH levels, signal-to-noise ratios, and the algorithms behind data compression. The inverse-of-exponential framing is what makes them everywhere.

What a Logarithm Is

The logarithm of $x$ to base $b$ is the exponent that turns $b$ into $x$:

$$\log_b(x) = y ;\iff; b^y = x.$$

Three conditions must hold: $b > 0$, $b \neq 1$, and $x > 0$.

The most common bases are:

• Common logarithm — $\log(x)$ usually means $\log_{10}(x)$.

• Natural logarithm — $\ln(x)$ means $\log_e(x)$, where $e \approx 2.71828$.

• Binary logarithm — $\log_2(x)$, used in computer science.

Quick facts.

• Definition: $\log_b(x) = y \iff b^y = x$.

• Domain: $x > 0$ (the input to log must be positive).

• Range: all real numbers.

• Special values: $\log_b(1) = 0$, $\log_b(b) = 1$.

• Cannot take log of: zero or negative numbers (over the real numbers).

• Grade introduced: CBSE Class 9–11 (logarithms appear in different chapters); CCSS-M HSF-LE.A.4 (for exponential models, express as a logarithm); NCERT Class 9 Chapter 1 — Number Systems (introductory log appears in advanced exercises).

The Seven Log Rules of Logarithms

1. Product rule

$\log_b(MN) = \log_b(M) + \log_b(N)$.

The log of a product is the sum of the logs.

2. Quotient rule

$\log_b(M/N) = \log_b(M) - \log_b(N)$.

The log of a quotient is the difference of the logs.

3. Power rule

$\log_b(M^p) = p \cdot \log_b(M)$.

The log of a power is the exponent times the log.

4. Change-of-base rule

$\log_b(x) = \dfrac{\log_c(x)}{\log_c(b)}$ for any base $c$.

Useful when your calculator only does $\log_{10}$ or $\ln$.

5. Identity rule

$\log_b(b) = 1$ and $\log_b(1) = 0$.

The log of the base is 1; the log of 1 is 0.

6. Inverse rules

$b^{\log_b(x)} = x$ and $\log_b(b^x) = x$.

Exponentiation and logarithm to the same base cancel.

7. Equal-arguments rule

If $\log_b(M) = \log_b(N)$, then $M = N$.

This is how log equations are solved.

Three Worked Examples of Logarithms

Quick. Evaluate $\log_2(8)$.

Ask: what power of 2 gives 8? $2^3 = 8$, so $\log_2(8) = 3$.

Final answer: $\log_2(8) = 3$.

Standard (Wrong Path First — Where Solutions Go Off the Rails). Solve $\log_3(x) + \log_3(x - 2) = 1$.

The wrong path. The rusher treats the left side term-by-term: $\log_3(x) = a$ and $\log_3(x - 2) = b$ with $a + b = 1$, and tries to solve two unknowns at once.

The rescue. Use the product rule.

$$\log_3(x) + \log_3(x - 2) = \log_3(x(x - 2)) = 1.$$

Convert to exponential form: $x(x - 2) = 3^1 = 3$.

Expand: $x^2 - 2x - 3 = 0$. Factor: $(x - 3)(x + 1) = 0$. So $x = 3$ or $x = -1$.

Check the domain: $\log_3(x)$ requires $x > 0$, and $\log_3(x - 2)$ requires $x > 2$. The value $x = -1$ violates both. Reject.

Final answer: $x = 3$.

Stretch. Solve $5^{x+1} = 2 \cdot 3^x$.

Take $\ln$ of both sides:

$$\ln(5^{x+1}) = \ln(2 \cdot 3^x).$$

Apply log rules:

$$(x + 1) \ln 5 = \ln 2 + x \ln 3.$$

$$x \ln 5 + \ln 5 = \ln 2 + x \ln 3.$$

$$x(\ln 5 - \ln 3) = \ln 2 - \ln 5.$$

$$x = \frac{\ln 2 - \ln 5}{\ln 5 - \ln 3} = \frac{\ln(2/5)}{\ln(5/3)}.$$

Numerically: $x \approx \dfrac{-0.916}{0.511} \approx -1.793$.

Final answer: $x = \ln(2/5) / \ln(5/3) \approx -1.79$.

Where Logarithms Show Up — From Earthquakes to Algorithms

Logarithms compress huge ranges into small ones. That single capability is why they appear in every quantitative field.

• Richter scale. Earthquake magnitude is $\log_{10}$ of the seismic amplitude. A magnitude-7 earthquake releases 10× the energy-density of a magnitude-6.

• Decibels. Sound intensity in decibels is $10 \log_{10}(I/I_0)$. The human ear's huge dynamic range collapses to 0–120 dB.

• pH scale. $\text{pH} = -\log_{10}[\text{H}^+]$. Hydrogen ion concentrations spanning 14 orders of magnitude become a 0–14 scale.

• Computer science. Binary search runs in $O(\log_2 n)$ time. The base-2 log is the number of yes/no questions to find one item among $n$.

• Information theory. Shannon entropy uses $\log_2$. The information content of a message is the log of its inverse probability.

• Compound interest. "How long to double money at 5%?" is $\log(2)/\log(1.05) \approx 14.2$ years.

The destination, in every direction: any time a quantity spans many orders of magnitude, logarithms compress it back into a useful range.

The Mistakes Students Make Most Often

1. Forgetting the domain

Where it slips in: Solving $\log_3(x) + \log_3(x - 2) = 1$ and accepting $x = -1$.

Don't do this: Skip the domain check.

The correct way: $\log_b(x)$ requires $x > 0$. Check every candidate solution against the original equation's domain restrictions.

2. Trying to take the log of a sum

Where it slips in: Writing $\log(a + b) = \log(a) + \log(b)$.

Don't do this: Distribute the log over addition.

The correct way: $\log(a + b)$ does not simplify in general. The product rule applies to products, not sums. $\log(ab) = \log(a) + \log(b)$.

3. Misapplying the power rule

Where it slips in: $\log(x^2) = (\log x)^2$.

Don't do this: Move the exponent inside as a power on the log.

The correct way: $\log(x^2) = 2 \log(x)$. The exponent comes out as a coefficient, not as an exponent on the log.

4. Trying to take the log of zero or a negative

Where it slips in: Computing $\log(0)$ or $\log(-5)$.

Don't do this: Treat these as defined.

The correct way: $\log(0)$ is undefined (the limit is $-\infty$). $\log$ of a negative number is undefined over the reals (defined in complex analysis but outside typical school context).

The real-world version. In 1935, Charles F. Richter at Caltech faced a measurement problem: earthquakes ranged across so many orders of magnitude that a linear scale was useless — a magnitude-2 quake was a hundredth of the magnitude-4. Richter took the logarithm of the seismic-wave amplitude and got the modern Richter scale, where 1–9 cover what would otherwise span ten million units.

The 1960 Valdivia earthquake (the largest ever recorded, magnitude 9.5) released about $10^{18}$ joules. The 2011 Tōhoku earthquake (magnitude 9.1) released about a third as much. Logarithms made these numbers comparable.

The Mathematicians Who Invented Logarithms

John Napier (1550–1617, Scotland) published Mirifici Logarithmorum Canonis Descriptio (1614), inventing logarithms. His goal was practical — reduce the labour of multiplying huge astronomical numbers. The breakthrough turned a 24-hour calculation into a 30-minute one.

Henry Briggs (1561–1630, England) worked with Napier to develop the base-10 (common) logarithms in 1617, leading to the publication of Arithmetica Logarithmica (1624) — the log tables that defined applied mathematics for the next 350 years.

Leonhard Euler (1707–1783, Switzerland) introduced the natural logarithm and the constant $e \approx 2.71828$ in Introductio in analysin infinitorum (1748), unifying logarithms with the exponential function and calculus.

The story worth remembering. Napier worked on logarithms for 20 years before publishing — partly because he was afraid the calculation might contain an error. Pierre-Simon Laplace later said Napier's invention "by shortening the labours doubled the life of the astronomer." Each Class 11 student uses Napier's invention every time a log button is pressed on a calculator.

Conclusion

• A logarithm $\log_b(x) = y$ is the exponent that satisfies $b^y = x$.

• The seven log rules — product, quotient, power, change-of-base, identity, inverse, equal-arguments — are the algebra of logs.

• The domain is $x > 0$; logs of zero or negative numbers are undefined over the reals.

• The single most common mistake is forgetting to check that candidate solutions to log equations satisfy the original domain.

• Logarithms compress huge ranges (earthquake magnitudes, decibels, pH) into useful scales.

Five Minutes of Practice

1. Evaluate $\log_5(125)$.

2. Solve $\log_2(x) + \log_2(x - 3) = 2$.

3. Solve $3^x = 10$ to two decimal places.

Want a live Bhanzu trainer to walk through more logarithm problems? Book a free demo class — online globally.