Question

lg(x+4)-lg(x-3)=lg 8 решите пожалуйста

Answer 1

ОДЗ:
$\left \{ {{x+4\ \textgreater \ 0} \atop {x-3\ \textgreater \ 0}} \right. \\ \left \{ {{x\ \textgreater \ -4} \atop {x\ \textgreater \ 3}} \right. \Rightarrow x\in(3;+\infty)$
Воспользуемся следующим свойством логарифмов:
$log_ab - log_ac = log_a \frac{b}{c} \\\\ lg(x+4)-lg(x-3)=lg8\\ lg \frac{x+4}{x-3}=lg8$
Справа и слева получили логарифм по одному и тому же основанию 10. Поэтому:
$\frac{x+4}{x-3}=8\\ x+4=8(x-3)\\ x+4=8x-24\\ x-8x=-24-4\\ -7x=-28 |:(-7)\\ x=4$

Полученный корень удовлетворяет ОДЗ.

Ответ: x=4

Answer 2

Solving the Equation lg(x+4)-lg(x-3)=lg8

To solve the equation lg(x+4)-lg(x-3)=lg8, we can use the properties of logarithms to simplify and solve for x.

First, let's use the property that lg(a) - lg(b) = lg(a/b). Applying this property to the given equation, we get:

lg(x+4) - lg(x-3) = lg8 lg((x+4)/(x-3)) = lg8

Now, we can use the property that lg(a) = b is equivalent to a = 10^b. Applying this property to the equation, we get:

(x+4)/(x-3) = 10^lg8 (x+4)/(x-3) = 8

To solve for x, we can cross multiply and solve for x:

(x+4) = 8(x-3) x+4 = 8x-24 4+24 = 8x-x 28 = 7x x = 28/7 x = 4

So, the solution to the equation lg(x+4)-lg(x-3)=lg8 is x = 4.