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
Evaluate $\log_5(125)$.
Solve $\log_2(x) + \log_2(x - 3) = 2$.
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.
Was this article helpful?
Your feedback helps us write better content