Answer: 1) Product property of logarithm,
2) Subtraction property of logarithm
3) Equality property of logarithm
Step-by-step explanation:
By the Product property of logarithm,
log a + log b = log(a.b)
And, By the Subtraction property of logarithm
log a - log b = log(a/b)
Also, by the equality property of logarithm,
log(a) = log(b) ⇒ a = b
Given expression,
[tex]log_2(5)+log_2(x+2)-log_2(x+1)=log_2(7)[/tex]
[tex]\implies log_2(5(x+2))-log_2(x+1))=log_2(7)[/tex] (Product property of logarithm)
[tex]\implies log_2(\frac{5(x+2)}{2(x+1)})=log_2(7)[/tex] (Subtraction property of logarithm)
[tex]\implies \frac{5(x+2)}{2(x+1)}=7[/tex] ( Equality property of logarithm )
[tex]\implies 5(x+2)=7(x+1)[/tex] ( Multiplicative property of equality )
[tex]\implies 5x+10=7x+7[/tex] ( Distributive property of equality )
[tex]\implies 3=2x[/tex] (subtraction property of equality )
[tex]x=\frac{3}{2}[/tex] (division property of equality )
