common logarithm

01 - Solution of Logarithmic Equations

Solve for x from the following:

  1. $\log_6 (x - 2) + \log_6 (x + 3) = 1$
     
  2. $x^{\log x} = 10\,000$

Logarithm and Other Important Properties in Algebra

Properties of Logarithm

  1. If   $y = a^x$,   then   $\log_a y = x$.   ← Definition of logarithm
  2. $\log_a xy = \log_a x + \log_a y$
  3. $\log_a \dfrac{x}{y} = \log_a x - \log_a y$
  4. $\log_a x^n = n \log_a x$
  5. $\log_a a = 1$
  6. $\log_a 1 = 0$
  7. $\log_{10} x = \log x$   ←   Common logarithm
  8. $\log_e x = \ln x$   ←   Naperian or natural logarithm
  9. $\log_y x = \dfrac{\log x}{\log y} = \dfrac{\ln x}{\ln y}$   ←   Change base rule
  10. If   $\log_a x = \log_a y$,   then   $x = y$.
  11. If   $\log_a x = y$,   then   $x = {\rm antilog}_a \, y$.

 

 
 
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