The inverse of exponential functions
Logarithms are the inverse of exponential functions. They "undo" exponentiation.
If b^y = x, then y = \log_b x
"y is the logarithm base b of x"
\log_{10} x or \log x
Base 10 logarithm
\ln x
Base e logarithm (e ≈ 2.718)
\log_2 x, \log_5 x, etc.
Any base greater than 0 and ≠ 1
If: =
Then:
If: log =
Then:
log₂ 8 = 3
Because 2³ = 8
log₁₀ 100 = 2
Because 10² = 100
ln e = 1
Because e¹ = e
These properties help simplify logarithmic expressions:
\log_b (MN) = \log_b M + \log_b N
Log of product = sum of logs
\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N
Log of quotient = difference of logs
\log_b (M^k) = k \log_b M
Log of power = exponent × log
\log_b M = \frac{\log_a M}{\log_a b}
Convert between bases
\log_b M = \log_b N \implies M = N
Equal logs imply equal arguments
\log_b 1 = 0
\log_b b = 1
Domain: M > 0, b > 0, b ≠ 1
1. log₂(8 × 4) = log₂8 + log₂4
Calculate: + =
2. log₁₀(1000) = 3 × log₁₀(10)
Calculate: 3 × =
Simplify using logarithm properties:
log₂ =
Logarithmic functions have distinct characteristics:
To solve logarithmic equations:
Step 1: Convert to exponential form
log₂(x + 3) = 4 → 2⁴ = x + 3
Step 2: Solve
16 = x + 3 → x = 13
Check: log₂(13 + 3) = log₂(16) = 4 ✓
Domain check: x + 3 = 16 > 0 ✓
Solve: log () =
1. Solve: log₂(x - 1) = 3
x =
2. Solve: ln(x + 5) = 2
x =
Logarithms appear in many scientific and real-world contexts:
pH = -log₁₀[H⁺]
Measures acidity of solutions
Magnitude = log₁₀(amplitude)
Measures earthquake intensity
dB = 10 log₁₀(I/I₀)
Measures sound loudness
log₂ n for algorithm complexity
Binary search, sorting algorithms