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Logarithms Worksheets

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Easy

10 problems

Medium

20 problems

Hard

20 problems

Mixed

30 problems

Free printable logarithms worksheets with step-by-step answer keys. Every worksheet is uniquely generated so students never see the same problems twice. Topics covered range from evaluate simple log, definition of logarithm at the easy level through to solve exponential equation using logs at the advanced level.

CCSS.HSF.BFCCSS.HSF.LE

What is logarithms?

A logarithm is the inverse operation of exponentiation, answering the question of what power a base must be raised to in order to produce a given result. The notation log_b(x) = n means that bn = x, where b is the base, x is the argument, and n is the result. For example, log_2(8) = 3 because 23 = 8.

Why it matters

Logarithms appear throughout science, engineering, and finance where exponential relationships dominate. The Richter scale uses base-10 logarithms to measure earthquake intensity, where each whole number represents a 10-fold increase in power. Sound intensity in decibels follows a logarithmic scale, making a 60-decibel conversation 1,000 times louder than a 30-decibel whisper. In finance, compound interest calculations often require logarithms to determine how long money takes to double or triple. Computer science relies heavily on base-2 logarithms for algorithm analysis, where log_2(1024) = 10 indicates that binary search can find any item among 1,024 elements in just 10 steps. The natural logarithm appears in calculus and serves as the foundation for exponential growth models in biology and economics. Students encounter logarithmic functions in Algebra II under CCSS.HSF.BF and CCSS.HSF.LE standards.

Common mistakes to watch for

  • ✗Confusing the base and argument positions, writing log_8(2) = 3 instead of log_2(8) = 3 when solving 2^3 = 8.
  • ✗Applying the product rule incorrectly by writing log_10(5 × 3) = log_10(5) × log_10(3) instead of log_10(5) + log_10(3).
  • ✗Forgetting to apply the power rule, calculating log_2(4^3) = log_2(64) = 6 instead of using log_2(4^3) = 3 × log_2(4) = 3 × 2 = 6.
  • ✗Mixing up logarithm properties when solving log_10(100/10) by writing log_10(100) + log_10(10) = 2 + 1 = 3 instead of log_10(100) - log_10(10) = 2 - 1 = 1.

Questions teachers ask

What is the difference between log and ln?+
The notation 'log' typically refers to the common logarithm (base 10), while 'ln' specifically means the natural logarithm (base e ≈ 2.718). Some contexts use 'log' to mean natural logarithm, so checking the base is important. Both follow the same rules and properties.
How do you check if a logarithm answer is correct?+
Convert the logarithm back to exponential form and verify. If log_3(27) = 3, check that 3^3 = 27. The base raised to your answer should equal the original argument. This method works for any logarithm problem.
Why can't you take the logarithm of a negative number?+
In real numbers, no positive base raised to any real power produces a negative result. Since logarithms ask 'what power gives this result?' and negative results are impossible with positive bases, logarithms of negative numbers are undefined in real number systems.
What does it mean when log_b(x) equals zero?+
When log_b(x) = 0, it means b^0 = x, and since any positive number raised to the zero power equals 1, this means x = 1. Therefore, the logarithm of 1 in any base always equals zero.
How are logarithms related to exponential equations?+
Logarithms are the key tool for solving exponential equations. When solving 2^x = 32, taking log_2 of both sides gives x = log_2(32) = 5. The logarithm isolates the variable from the exponent position, making exponential equations solvable.
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Beginner

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Concepts
Evaluate simple log, definition of logarithm
Range
base 2/3/5/10, exp 1–3
Steps
3 steps
Example
log₂(8) = ?

Easy

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Concepts
Evaluate log with higher exponents
Range
base 2/3/5/10, exp 2–5
Steps
4 steps
Example
log₃(243) = ?

Medium

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Concepts
Log rules: product, quotient, power
Range
base 2/10, values are perfect powers
Steps
4 steps
Example
log₂(4 × 8) = ?

Hard

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Concepts
Solve exponential equation using logs
Range
base 2/3/5, exponent 2–5
Steps
5 steps
Example
3^x = 243

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