Logarithms
Understand and apply logarithmic functions and properties
Start with the basics and progress through 12 lessons. Each lesson builds on the previous one.
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10 questions, new mix each time (from 213)
In This Topic (12 lessons)
Introduction to Logarithms
Learn what logarithms are and how they relate to exponents as inverse operations.
Definition of a Logarithm
Understand what logarithms are and how they relate to exponents as inverse operations.
Common Logarithms (Base 10)
Learn the common logarithm log x with base 10: exact values for powers of 10, estimating logs between whole numbers, and using a calculator.
Natural Logarithms (ln)
Learn what ln x means, why ln e = 1 and ln 1 = 0, how ln undoes e^x, and how to solve equations like e^x = 20 with a calculator.
Graphing Logarithmic Functions
Learn to graph logarithmic functions: key points, domain, range, the vertical asymptote, the mirror image of the exponential function and shifts.
Properties of Logarithms
Learn the basic properties of logarithms: the log of 1, the log of the base, the two inverse properties and the one-to-one property.
Product Rule for Logarithms
Learn why the logarithm of a product equals the sum of the logarithms, and use it to expand, condense and evaluate logs.
Quotient Rule for Logarithms
Learn why the logarithm of a quotient equals a difference of logarithms, and use the rule to expand and condense expressions.
Power Rule for Logarithms
Learn how the power rule moves an exponent in front of a logarithm, how roots become fractional powers, and how to expand and condense logs with all three rules.
Change of Base Formula
Use the change of base formula to evaluate a logarithm of any base with the log or ln key of your calculator.
Solving Logarithmic Equations
Solve equations with the unknown inside a logarithm, reject extraneous solutions, and use logarithms to solve exponential equations like 3^x = 20.
Logarithm Word Problems
Use logarithms to solve real problems: pH, decibels, earthquake magnitudes, doubling time, half-life and how long money takes to grow.
Logarithms are the inverse of exponential functions, answering the question: to what power must the base be raised to get a certain value? Essential for solving exponential equations, logarithms appear in sciences measuring orders of magnitude.
What You'll Learn
- Convert between exponential and logarithmic forms
- Evaluate logarithms without a calculator
- Apply logarithm properties
- Solve logarithmic equations
- Use logarithms in real-world applications
Frequently Asked Questions
What does a logarithm mean?
$\log_b(x) = y$ means $b^y = x$. For example, $\log_2(8) = 3$ because $2^3 = 8$.
What are the main logarithm properties?
Product: $\log(ab) = \log(a) + \log(b)$. Quotient: $\log(a/b) = \log(a) - \log(b)$. Power: $\log(a^n) = n\log(a)$.
What is the difference between log and ln?
log usually means base 10 (common log). ln means base e (natural log). Both follow the same properties.