Question

Lg(2x - 1) + Ig(x - 9) = 2

Answer 1

Ответ:

х=13

Объяснение:

1) область определения:

{ 2x−1 > 0,

{ x−9 > 0

⇒ x>9

2) сумма логарифмов равна логарифму произведения:

lg((2x−1)(x−9)) = lg 100

(2x−1)(x−9) = 100

2x² − 19x + 9 − 100 = 0

2x² − 19x − 91 = 0

(x−13)(2x+7) = 0

Поскольку из п. 1 x>9, то единственное решение x=13

ОТВЕТ: x=13

Answer 2

Ответ:

$13$

Объяснение:

ОДЗ:

$\left \{ {{2x-1 > 0} \atop {x-9 > 0}} \right. \Leftrightarrow \left \{ {{2x > 1} \atop {x > 9}} \right. \Leftrightarrow \left \{ {{x > 0,5} \atop {x > 9}} \right. \Leftrightarrow x > 9;$

Решение:

$lg(2x-1)+lg(x-9)=2;$

$lg((2x-1)(x-9))=lg(100);$

$(2x-1)(x-9)=100;$

$2x^{2}-18x-x+9-100=0;$

$2x^{2}-19x-91=0;$

$D=b^{2}-4ac;$

$D=(-19)^{2}-4 \cdot 2 \cdot (-91)=361+364 \cdot 2=361+728=1089;$

$1089=3 \cdot 363=3 \cdot 3 \cdot 121=3^{2} \cdot 11^{2}=(3 \cdot 11)^{2}=33^{2};$

$x_{1,2}=\frac{-b \pm \sqrt{D}}{2a};$

$x_{1}=\frac{-(-19)+\sqrt{33^{2}}}{2 \cdot 2}=\frac{19+33}{4}=\frac{52}{4}=13;$

$x_{2}=\frac{-(-19)-\sqrt{33^{2}}}{2 \cdot 2}=\frac{19-33}{4}=\frac{-14}{4}=-3,5;$

Корень x₂ не удовлетворяет ОДЗ.

Answer 3

It looks like you have an equation involving logarithmic and inverse trigonometric functions:

lg(2x - 1) + Ig(x - 9) = 2

I'm assuming that "lg" represents the base-10 logarithm (common logarithm), and "Ig" represents the natural logarithm (base e, also denoted as "ln").

To solve this equation, you can follow these steps:

Step 1: Combine the logarithmic terms. lg(2x - 1) + ln(x - 9) = 2

Step 2: Isolate one of the logarithmic terms. Let's isolate the natural logarithm term (ln):

ln(x - 9) = 2 - lg(2x - 1)

Step 3: Take the exponential of both sides to eliminate the natural logarithm: x - 9 = e^(2 - lg(2x - 1))

Step 4: Simplify the right side of the equation by using properties of logarithms and exponentials: x - 9 = e^2 / (2x - 1)

Step 5: Multiply both sides by (2x - 1) to get rid of the fraction: x(2x - 1) - 9(2x - 1) = e^2

Step 6: Expand and simplify the equation: 2x^2 - x - 18x + 9 = e^2

Step 7: Combine like terms: 2x^2 - 19x - 9 = e^2

Step 8: Move all terms to one side to set the equation to zero: 2x^2 - 19x - 9 - e^2 = 0

Now, this is a quadratic equation in terms of 'x'. You can use the quadratic formula to solve for 'x':

x = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 2, b = -19, and c = -9 - e^2. Plug these values into the quadratic formula:

x = (-(-19) ± √((-19)² - 4(2)(-9 - e^2))) / (2(2))

x = (19 ± √(361 - 4(2)(-9 - e^2))) / 4

x = (19 ± √(361 + 72 + 4e^2)) / 4

x = (19 ± √(433 + 4e^2)) / 4

So, the solutions for 'x' are given by: x₁ = (19 + √(433 + 4e^2)) / 4 x₂ = (19 - √(433 + 4e^2)) / 4

These are the solutions to the equation lg(2x - 1) + ln(x - 9) = 2.