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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