01 Key Concepts
Logarithm Definition
log_b(x) = y means the same thing as b^y = x — 'the log base b of x is y'.
Common Log and Natural Log
log(x) usually means log base 10. ln(x) means log base e (the natural logarithm).
Product Rule
log_b(m*n) = log_b(m) + log_b(n) — logs turn multiplication into addition.
Quotient Rule
log_b(m/n) = log_b(m) - log_b(n) — logs turn division into subtraction.
Power Rule
log_b(m^n) = n * log_b(m) — logs turn exponents into multiplication out front.
02 Key Formulas
- log_b(x) = y means b^y = x
- log_b(m*n) = log_b(m) + log_b(n)
- log_b(m/n) = log_b(m) - log_b(n)
- log_b(m^n) = n * log_b(m)
03 Solved Examples
Example 1 Evaluate log_2(8).
- Ask: 2 raised to what power gives 8?
- 2^3 = 8.
Answer: log_2(8) = 3
Example 2 Evaluate log_10(1000).
- Ask: 10 raised to what power gives 1000?
- 10^3 = 1000.
Answer: log_10(1000) = 3
Example 3 Use the product rule to simplify log_2(4) + log_2(8).
- Apply the product rule in reverse: log_2(4)+log_2(8) = log_2(4*8) = log_2(32).
- Ask: 2 raised to what power gives 32?
- 2^5 = 32.
Answer: log_2(32) = 5
04 Practice Questions
1Evaluate log_2(16).
4
2Evaluate log_10(100).
2
3Evaluate log_3(9).
2
4Use the product rule: log_2(2) + log_2(16).
log_2(32) = 5
5What does log_b(x)=y mean in exponential form?
b^y = x
📄 Logarithms — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.