text: # LOgarithm proBLeM
  
  • Nov 29th 2008, 05:07 AM
  mathbuoy
  LOgarithm proBLeM
  If log[a^2]b^2 - log[b]a = 4, find log[a]b.
  
  #characters in [ ] is at lower 'case' , attached to 'log'.
  
  *show working and explanation, pls...(Doh)
  • Nov 29th 2008, 05:30 AM
  running-gag
  Use the formula
  $\displaystyle log_a(x)\;=\;\frac{ln(x)}{ln(a)}$
  • Nov 29th 2008, 02:28 PM
  Soroban
  Hello, mathbuoy!
  
  This is a strange one . . . I don't recommend natural logs.
  
  Quote:
  
  If $\displaystyle \log_{a^2}(b^2) - \log_b(a) \:=\: 4$, find $\displaystyle \log_a(b)$
  This is what I do when I encounter a strange base . . .
  
  Let $\displaystyle \log_{a^2}(b^2) \:=\:p \quad\Rightarrow\quad(a^2)^p = b^2 \quad\Rightarrow\quad a^{2p} = b^2$
  
  Take logs (base $\displaystyle a$): .$\displaystyle \log_a(a^{2p}) \:=\:\log_a(b^2) \quad\Rightarrow\quad 2p\log_a(a) \:=\:2\log_a(b) \quad\Rightarrow\quad p \:=\:\log_a(b)$
  
  . . Hence: .$\displaystyle \log_{a^2}(b^2) \;=\;\log_a(b)$
  
  And equation becomes: .$\displaystyle \log_a(b) - \log_b(a) \;=\;4$
  
  Use this identity on the second term: .$\displaystyle \log_b(a) \:=\:\frac{1}{\log_a(b)}$
  . . and we have: .$\displaystyle \log_a(b) - \frac{1}{\log_a(b)} \;=\;4$
  
  Multiply by $\displaystyle \log_a(b)\!:\quad \left[\log_a(b)\right]^2 - 1 \;=\;4\log_a(b) \quad\Rightarrow\quad\left[\log_a(b)\right]^2 - 4\log_a(b) - 1 \;=\;0$
  
  Quadratic Formula: .$\displaystyle \log_a(b) \;=\;\frac{4 \pm\sqrt{20}}{2}$
  
  . . Therefore: .$\displaystyle \log_a(b)\;=\;2 \pm\sqrt{5}$
  
  • Nov 30th 2008, 04:50 AM
  running-gag
  Quote:
  
  Originally Posted by Soroban
  Hello, mathbuoy!
  
  This is a strange one . . . I don't recommend natural logs.
  
  It works well anyway !
  We have to find $\displaystyle X\;=\;log_a(b)\;=\;\frac{ln(b)}{ln(a)}$
  
  $\displaystyle log_{a^2}(b^2)\;=\;\frac{ln(b^2)}{ln(a^2)}$
  
  $\displaystyle log_{a^2}(b^2)\;=\;\frac{2ln(b)}{2ln(a)}$
  
  $\displaystyle log_{a^2}(b^2)\;=\;\frac{ln(b)}{ln(a)}$
  
  $\displaystyle log_{a^2}(b^2)\;=\;X$
  
  Equation becomes
  X - 1/X = 4
  X² - 4X - 1 = 0