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lorasvet [3.4K]
2 years ago
12

Solve for x : log(3x) + log(x + 4) = log(15).Please explain it to me.​

Mathematics
2 answers:
Dmitry [639]2 years ago
8 0

Answer:

x=1

Step-by-step explanation:

Given:

\log (3x)+\log(x+4)=\log (15)

As the logs have no base, assume that the base is 10.

\textsf{Apply log Product law}: \quad \log_ax + \log_ay=\log_axy

\implies \log_{10} (3x)+\log_{10}(x+4)=\log_{10} (15)

\implies \log_{10} \left(3x(x+4)\right)=\log_{10} (15)

Expand the brackets:

\implies \log_{10} \left(3x^2+12x\right)=\log_{10} (15)

\textsf{Apply the log Equality law}: \quad \textsf{if }\: \log_ax=\log_ay\:\textsf{ then }\:x=y

\implies 3x^2+12x=15

Subtract 15 from both sides:

\implies 3x^2+12x-15=0

Factor out the common term 3:

\implies 3(x^2+4x-5)=0

Divide both sides by 3:

\implies x^2+4x-5=0

Split the middle term:

\implies x^2+5x-x-5=0

Factorize the first two terms and the last two terms separately:

\implies x(x+5)-1(x+5)=0

Factor out the common term (x + 5):

\implies (x-1)(x+5)=0

Therefore:

x-1=0 \implies x=1

x+5=0 \implies x=-5

As logs cannot be taken of negative numbers, x=-5 is an <u>extraneous solution</u>.  Therefore, the only <u>valid solution</u> is:  x=1

zlopas [31]2 years ago
3 0

Answer:

x = 1

Explanation:

\sf \rightarrow log(3x) + log(x + 4) = log(15)

Rule: log(a) + log(b) = log(ab)

\sf \rightarrow log(3x(x + 4)) =  log(15)

cancel out log on both sides

\sf \rightarrow 3x(x + 4) = 15

relocate constant variable

\sf \rightarrow 3x^2 + 12x -15 = 0

take 3 as a common factor

\sf \rightarrow 3(x^2 + 4x -5) = 0

divide both sides by 3

\sf \rightarrow x^2 + 4x -5 = 0

middle term split

\sf \rightarrow x^2 + 5x -x-5 = 0

factor common terms

\sf \rightarrow x(x  + 5) -1(x+5)= 0

collect into groups

\sf \rightarrow (x-1)(x+5)= 0

set to zero

\sf \rightarrow x-1 = 0 , \ x+5= 0

relocate variables

\sf \rightarrow x = 1,  \ x = -5

There must be a positive solution for log, so the solution is only x = 1

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