SOLUTION: How do I solve these problems? 1. log(2 - x) = 0.5 2. log(x + 25) = log(x + 10) + log 4

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Question 965036: How do I solve these problems?
1. log(2 - x) = 0.5
2. log(x + 25) = log(x + 10) + log 4

Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
the first one is solved as follows:

log(2-x) = .5

this is true if and only 10^.5 = 2-x

square both sides of this equation to get:

10 = (2-x)^2

simplify to get:

10 = 4 - 4x + x^2

subtract 10 from both sides and re-arrange the terms to get:

x^2 - 4x - 6 = 0

factor using the quadratic formula to get:

x = 5.16227766

x = -1.16227766

both values work.

just replace them in the original equation and you'll see.

the second problem is solved as follows:

log(x + 25) = log(x + 10) + log 4

this is equivalent to log(x + 25) = log(4 * (x + 10) which is equivalent to log(x + 25) = log(4x + 40).

this is true if and only if x + 25 = 4x + 40

solve for x to get x = -5.

replace x with -5 in the original equation and you'll see that they're equivalent.

log(-5 + 25) = log(20).\

log(x + 10) + log(4) is equal to log(5) + log(4) which is equal to log(5 * 4) which is equal to log(20).

the rule of logs that are used are:

logb(x) = y if and only if b^y = x

logb(a) + logb(b) = logb(a * b)