🔢 Exponents & Logarithmic Scales
Logarithm & Change of Base Calculator
Evaluate logarithms of any arbitrary base ($\log_b(x) = \frac{\ln x}{\ln b}$), convert to natural ($\ln$) and binary ($\log_2$) bases, and calculate inverse anti-logarithms ($b^y$).
🔟 Base 10 ($\log_{10}$)
🌿 Base $e$ ($\ln$)
💻 Base 2 ($\log_2$)
🔢 Base 16 ($\log_{16}$)
Logarithm Inputs
Calculated Value log₁₀(1000)
3.0000
Natural Log ($\ln x$)
6.9078
Binary Log ($\log_2 x$)
9.9658
Common Log ($\log_{10} x$)
3.0000
💡 Change of Base Theorem:
$$\log_{10}(1000) = \frac{\ln(1000)}{\ln(10)} = \frac{6.907755}{2.302585} = 3.00000$$
Logarithm Laws & Algebraic Identities
Logarithms represent the inverse operation to exponentiation ($b^y = x \iff \log_b(x) = y$):
- Change of Base Rule: $$\log_b(x) = \frac{\ln(x)}{\ln(b)} = \frac{\log_{10}(x)}{\log_{10}(b)}$$
- Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$
- Quotient Rule: $\log_b(x/y) = \log_b(x) - \log_b(y)$
- Power Rule: $\log_b(x^k) = k \log_b(x)$