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log x + log (x-3) = 1

Sagot :

Given:

The logarithmic equation is:

[tex]\log x+\log (x-3)=1[/tex]

To find:

The value of x.

Solution:

Properties of logarithm used:

[tex]\log a+\log b=\log (ab)[/tex]

[tex]\log 10=1[/tex]

[tex]\log x[/tex] is defined for [tex]x>0[/tex].

We have,

[tex]\log x+\log (x-3)=1[/tex]

Using properties of logarithm, we get

[tex]\log [x(x-3)]=\log 10[/tex]

[tex]x^2-3x=10[/tex]

Splitting the middle term, we get

[tex]x^2-5x+2x-10=0[/tex]

[tex]x(x-5)+2(x-5)=0[/tex]

[tex](x-5)(x+2)=0[/tex]

[tex]x=5,-2[/tex]

In the given equation, we have a term [tex]\log x[/tex]. It means the value of x must be greater than 0 or positive. So, the only possible value of x is:

[tex]x=5[/tex]

Therefore, the value of x is 5.

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