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For what value of x is the following true? log(x+9)=logx+log9.

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3 Answers

log(xy) = log x + log y
~~~~~~~~  

log(x + 9) = log x + log 9

log(x + 9) = log 9x

9x = x + 9 

8x = 9

x = 9/8
log(x+9)=log(x)+log(9)
 
When solving logarithmic equations, you always want to put your logarithms together if their is an addition sign between two terms. For this, you use the properties of logarithms as such:
 
log(x+9)=log(9x)   ==>  [ from log(a)+log(b)=log(ab) ]
 
Now you exponentiate both sides of the equation and get:
 
x+9=9x
 
8x=9
 
x=9/8
 
 
log(x+9) = log x + log 9
 
Since log(a) + log(b) = log(ab), 
 
log(x+9) = log (x*9)
 
 
 
Rewrite the x*9 as 9x:
 
log (x+9) = log (9x)
 
 
 
Raise 10 to both sides, which is the inverse of the log function, resulting in:
 
x+9 = 9x
 
Subtract x from both sides:
 
9 = 8x
 
Divide both sides by 8:
x = 9/8
 
 
Check:
 
log ((9/8) + 9) = log (9/8) + log 9
 
log (81/8) = log ((9/8)(9)) = log (81/8)