July 24, 2026
How to Understand Logarithms: Rules and Examples
Logarithms explained: what a log means, the product, quotient, and power rules, the change of base formula, and how logs connect to exponents, with examples.
The direct answer: a logarithm answers the question "what exponent gives this result," so log base b of x equals y means b to the y equals x, and it follows three working rules (product, quotient, power) plus a change of base formula. This guide builds the rules from exponents with examples you can copy.
Logarithms at a Glance
| Question | Answer |
|---|---|
| What is log base b of x? | The exponent y where b to the y = x. |
| What is log of 1? | 0, because any base to the 0 is 1. |
| What is log of the base? | 1, because b to the 1 = b. |
| What are the three rules? | Product, quotient, and power. |
| How do I compute odd bases? | Use the change of base formula. |
What a Logarithm Means
The definition is the key. log base b of x = y if and only if b to the y = x, with b greater than 0, b not equal to 1, and x greater than 0.
Worked examples
log base 2 of 8 = 3 because 2 cubed = 8. log base 10 of 1000 = 3 because 10 cubed = 1000.
Common log, written log x, means base 10. Natural log, written ln x, means base e, where e is about 2.718.
Two values appear constantly: log base b of 1 = 0 and log base b of b = 1. Also b to the (log base b of x) = x, which shows logs and exponentials undo each other. Khan Academy frames logs as the inverse of exponentials with the same set of examples Khan Academy.
The Product Rule
The logarithm of a product equals the sum of the logarithms:
log base b of (M times N) = log base b of M plus log base b of N
Worked example
log base 2 of (8 times 4) = log base 2 of 8 plus log base 2 of 4 = 3 + 2 = 5. Check: 8 times 4 = 32 = 2 to the 5, correct.
The Quotient Rule
The logarithm of a quotient equals the difference:
log base b of (M divided by N) = log base b of M minus log base b of N
Worked example
log base 3 of (81 divided by 9) = log base 3 of 81 minus log base 3 of 9 = 4 minus 2 = 2. Check: 81 divided by 9 = 9 = 3 squared, correct.
The Power Rule
An exponent inside the log becomes a multiplier outside:
log base b of (M to the p) = p times log base b of M
This holds for any real exponent p, including fractions and negatives.
Worked example
log base 2 of (8 to the 4) = 4 times log base 2 of 8 = 4 times 3 = 12.
A useful special case: log base b of the square root of M = one half times log base b of M.
Change of Base
To compute a log in any base on a calculator, rewrite it using base 10 or base e:
log base b of x = log base c of x divided by log base c of b
Worked example
log base 2 of 10 = ln 10 divided by ln 2, which is about 2.3026 divided by 0.6931, about 3.322.
A neat fact: log base a of b = 1 divided by log base b of a.
Watch the Fake Rules
These look tempting but are wrong:
- log(a plus b) is not log a plus log b.
- log(a minus b) is not log a minus log b.
- (log a) squared is not 2 log a.
Sums and differences inside a log cannot be split. The product rule only applies to multiplication inside the argument.
Logs in the Real World
The rules above are not just algebra drills. Log scales turn multiplication into addition, which is why they describe quantities that span huge ranges.
Decibels and sound
Sound intensity in decibels is 10 times log base 10 of (intensity divided by reference intensity). Because the scale is logarithmic, adding 10 decibels means multiplying the actual intensity by 10, not adding a fixed amount. A whisper and a jet engine differ by many orders of magnitude, and a log scale is the only way to plot both on one axis without crushing the small one.
pH in chemistry
pH is minus log base 10 of the hydrogen ion concentration. A drop from pH 7 to pH 6 means ten times more hydrogen ions, not one unit more. This is why small pH shifts in a lake or a blood sample signal large chemical changes, and why the power rule matters: log of a concentration ratio becomes a difference you can read directly.
Growth and decay
Because logs invert exponentials, they turn an exponential growth curve into a straight line when you plot the log of the value. Fitting a line to that plot is how scientists estimate a growth rate from messy data. The connection back to exponents is what makes the whole topic click.
Logs of Powers of the Base
A fast shortcut: log base b of (b to the k) = k, because the log asks "what exponent gives this," and the answer is sitting there. So log base 2 of 32 = 5, since 32 = 2 to the 5, and ln of e cubed = 3. This is the inverse property written as a fact, and it lets you estimate logs without a calculator.
Estimating without a calculator
To place log base 10 of 500, note 500 is between 100 = 10 squared and 1000 = 10 cubed, so the log is between 2 and 3. Since 500 is past the midpoint on a log scale (which is about 316 = 10 to the 2.5), the value is a bit above 2.7. The exact figure is about 2.699. Estimation like this catches typos in calculator output, which is the habit that keeps your answers honest on exams.
The Number e and the Natural Log
The natural log uses base e, about 2.718, a number that appears wherever change is continuous. If you earn 100 percent interest compounded continuously, each dollar becomes e dollars after a year. If a population grows at a fixed percent rate, its size follows e to the (rate times time). The natural log is the inverse of that growth, so ln answers "how long" given "how much."
Why e, not just 10
Base 10 logs are convenient for orders of magnitude. Base e is convenient for calculus, because the function e to the x is its own derivative, the only function with that property. That single fact is why ln and e show up together in every later course from physics to finance, and why learning the natural log now pays off long after the algebra test.
Common Misconceptions
- Forgetting logs need a positive argument.
- Applying the product rule to a sum inside the log.
- Mixing up the order in the quotient rule.
- Using the power rule on the whole log instead of the argument.
- Dropping the base when it matters.
- Thinking ln and log are interchangeable. They use different bases and give different numbers.
- Believing a log is always positive. When the argument is between 0 and 1, the value is negative.
Frequently Asked Questions
Are logs and exponents opposites?
Yes. A logarithm undoes an exponential, which is why b to the (log base b of x) = x.
When would I use a logarithm in real life?
For measuring sound in decibels, earthquake magnitude, and pH in chemistry, all of which use log scales. The Richter scale and decibel scale compress huge ranges into readable numbers.
Why is the natural log special?
Its base e appears in growth, decay, and calculus, so ln shows up often in later math.
Can a log be negative?
Yes, when the argument is between 0 and 1, the exponent needed is negative.
How do I solve a log equation?
Use the rules to combine terms, then rewrite in exponential form and solve.
How do logs connect to exponents and functions?
Logs reverse exponents, and log functions such as f(x) = ln x are a standard family you graph in precalculus.
Sources
About the author
Michael R. is a study skills coach with 12 years of experience and a learning specialist. He helps students develop effective study strategies and organizational systems.