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Learn Logarithms

A logarithm answers one question: what exponent do I need?

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A logarithm is an exponent in disguise. Log base 2 of 8 equals 3 because 2 to the third power is 8, so every log statement is a rewritten exponential statement. Logarithms turn multiplication into addition, which is why they were used for centuries to speed up calculation, and why they appear in decibels, the pH scale and earthquake magnitude scales. A course on logs covers converting between forms, the product, quotient and power rules, the natural log ln with base e, change of base and solving equations where the unknown sits in an exponent. Learnr builds a focused logarithms course and teaches it in short interactive lessons with instant feedback.

How to learn Logarithms, step by step

Translate every log into an exponent question. Read log base 5 of 25 as "5 to what power gives 25?" The answer is 2. Practicing this translation until it is automatic makes every later rule feel natural.
Evaluate simple logs mentally. Work out log base 2 of 32, log base 10 of 1000 and log base 3 of 1/9, which are 5, 3 and -2. Note that log of 1 is always 0 and the log of a negative number is undefined.
Learn the three rules and why they work. Log of ab is log a plus log b, log of a over b is log a minus log b, and log of a to the n is n log a. Each comes straight from an exponent rule, so derive them once.
Meet e and the natural log. ln x is log base e, where e is about 2.718. It is the log of continuous growth, so ln appears in compound interest, population models and calculus.
Solve equations with the unknown in an exponent. For 2 to the x equals 10, take ln of both sides to get x ln 2 = ln 10, so x is ln 10 divided by ln 2, about 3.32. Check that 2 cubed is 8 and 2 to the fourth is 16, so the answer is reasonable.
Review the rules in short daily sessions. Log rules blur together without regular use. Five minutes a day in a Learnr course, kept going by a streak, keeps them distinct and catches the classic errors early.

What you'll learn

Convert between logarithmic and exponential form
Evaluate logs like log base 3 of 81 without a calculator
Apply the product, quotient and power rules correctly
Use natural logs and the change-of-base formula
Solve exponential and logarithmic equations, including growth and decay problems

What's inside your Logarithms course

A sample path Learnr builds for Logarithms. Your real course adapts to your goal and level.

1. Exponents review and the meaning of a logarithm
2. Evaluating logs and common bases
3. Product, quotient and power rules
4. Natural logarithms and the number e
5. Change of base and solving exponential equations
6. Log equations, graphs and real applications

How long does it take to learn Logarithms?

Most learners get comfortable with logarithms in about 2 to 4 weeks at 3 to 4 hours a week, assuming exponent rules are solid. Applied growth and decay problems take an extra week or two.

What you need before you start

Exponent rules, including negative and fractional exponents, plus algebra for solving equations. If exponents feel shaky, review them first, since logs are built on them.

Common mistakes to avoid

Splitting the log of a sum: log(a + b) is not log a plus log b. The product rule only applies to the log of a product.
Cancelling log from a fraction, as if log 8 over log 2 equals 8 over 2. It actually equals 3, which is log base 2 of 8.
Accepting solutions that make a log's input zero or negative. Always check answers in the original equation.

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Logarithms questions, answered

What is the difference between log and ln?

ln means the natural log, with base e (about 2.718). Log with no base written usually means base 10 on calculators and in school, though some higher math and programming languages use log for the natural log.

Why do we need logarithms?

They let you solve for an unknown exponent, compress huge ranges of values into readable scales like decibels and pH, and turn products into sums, which simplifies many calculations in science and finance.