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Logarithms

The logarithm is the inverse operation of a power: log⁡bx=y\log_b x=y means by=xb^y=x. We use logarithms to solve equations where the unknown is in the exponent.

log⁡bx=y  ⟺  by=x\log_b x = y \iff b^y = xdefinition of a logarithm
ln⁡(ab)=ln⁡a+ln⁡b,ln⁡(an)=nln⁡a\ln(ab) = \ln a + \ln b,\quad \ln(a^n) = n\ln alogarithm rules

Symbols

bbbase
xxargument, x>0x>0

Example

Solve 1.05t=1.51.05^t = 1.5: t=ln⁡1.5ln⁡1.05≈8.31t = \dfrac{\ln 1.5}{\ln 1.05} \approx 8.31.

ln⁡(a+b)\ln(a+b) is not ln⁡a+ln⁡b\ln a + \ln b - the logarithm of a sum cannot be simplified that way.
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Part of Foundations of Mathematics: Exponential and logarithmic functions.