# Simpel Log

• Sep 14th 2009, 02:27 AM
sf1903
Simpel Log
log3(x+1)=log9(26+2x)
X?

with calculation...
• Sep 14th 2009, 02:59 AM
Amer
Quote:

Originally Posted by sf1903
log3(x+1)=log9(26+2x)
X?

with calculation...

$\displaystyle \log _{3} (x+1) = \log _{9} (26+2x)$

$\displaystyle 26+2x = 9^{\log_3 (x+1)}$

$\displaystyle 26+2x = (3^{\log_3 (x+1)})^2$

$\displaystyle 26+2x = (x+1)^2$ you can continue from here
• Sep 14th 2009, 06:43 AM
pacman
we may use this logarithmic identity by EULER, log (base b) a = (log a)/(log b)

solve for x: log3(x+1)=log9(26+2x)

-------------------------------------------------------

log (x + 1)/(log 3) = log (2x + 26)/(log 9) = log (2x + 26)/(2 log 3)

cross-multiply,

2 (log 3) log (x + 1) = (log 3) log (2x + 26), cancel log 3

2 log (x + 1) = log (2x + 26),

log (x + 1)^2 = log (2x + 26),

take anti-log og both sides,

(x + 1)^2 = 2x + 26,

x^2 + 2x + 1 = 2x + 26,

2x^2 - 2x + 2x - 26 +1 = 0,

x^2 - 25 = 0

factoring,

(x + 5)(x - 5) =0,

x = {-5, 5}, but only 5 is the valid answer

the graph below is wrong
• Sep 14th 2009, 06:53 AM
pacman
x^2 - 25 = 0, this is the graph
• Sep 14th 2009, 08:37 AM
Soroban
Hello, sf1903!

Another approach . . .

Quote:

Solve for $\displaystyle x\!:\;\;\log_3(x+1)\:=\:\log_9(2x+26)$

Let: .$\displaystyle \log_3(x+1) \:=\:P \quad\Rightarrow\quad 3^P \:=\:x+1$

Square both sides: .$\displaystyle (3^P)^2 \:=\:(x+1)^2 \quad\Rightarrow\quad (3^2)^P \:=\:(x+1)^2 \quad\Rightarrow\quad 9^P \:=\:(x+1)^2 \quad\Rightarrow\quad P \:=\:\log_9(x+1)^2$

. . We have: .$\displaystyle \log_3(x+1) \:=\:\log_9(x+1)^2$

The equation becomes: .$\displaystyle \log_9(x+1)^2 \:=\:\log_9(2x+26)$

Exponentiate both sides: .$\displaystyle (x+1)^2 \:=\:2x + 26$

. . and so on . . .