Question

Log3 log3 x=log9 (5-4log3 x)

Answer 1

Решение во вложении----------------

Answer 2

$log_3(log_3x)=log_9(5-4log_3x)$

ОДЗ: 
$\left[\begin{array}{ccc}log_3x\ \textgreater \ 0\\5-4log_3x\ \textgreater \ 0\\x\ \textgreater \ 0\end{array}\right\left[\begin{array}{ccc}log_3x\ \textgreater \ log_33^0\\5\ \textgreater \ 4log_3x\\x\ \textgreater \ 0\end{array}\right\left[\begin{array}{ccc}x\ \textgreater \ 3^0\\\frac{5}{4}\ \textgreater \ log_3x\\x\ \textgreater \ 0\end{array}\right\\\\\\\left[\begin{array}{ccc}x\ \textgreater \ 1\\log_33^\frac{5}{4}\ \textgreater \ log_3x\end{array}\right\left[\begin{array}{ccc}x\ \textgreater \ 1\\3^{\frac{5}{4}}\ \textgreater \ x\end{array}\right lODZ:1\ \textless \ x\ \textless \ 3^{\frac{5}{4}}$

по свойствам логарифмов: 
$log_3(log_3x)=log_9(5-4log_3x)\\log_3(log_3x)=log_{3^2}(5-4log_3x)=log_3\sqrt{5-4log_3x}\\log_3x=\sqrt{5-4log_3x}$

возведём в квадрат обе части уравнения, чтобы избавиться от иррациональности: 
$(log_3x)^2=5-4log_3x$

обозначим $log_3x$ переменной $a$, тогда 
$a^2=5-4a\to a^2+4a-5=0\\D=16+20=36=6^2\\a_1=\frac{-4+6}{2}=1\\a_2=\frac{-4-6}{2}=-5$

из вышеприведённой замены: 
$\left[\begin{array}{ccc}log_3x=1\\log_3x=-5\end{array}\right\left[\begin{array}{ccc}x=3^1\\x=3^{-5}\end{array}\right$

$3^{-5}$ меньше единицы, а потому является исключением ОДЗ, определённого ранее; ответ: $x=3$

Answer 3

To solve the equation log3 + log3(x) = log9 + (5 - 4log3 + x), we can simplify the equation using logarithmic properties and then solve for x.

Step 1: Simplify the equation using logarithmic properties: Since loga(x) + loga(y) = loga(xy), we can simplify the left side of the equation: log3(x) + log3(x) = log3(x * x) = log3(x^2)

Step 2: Apply the logarithmic property to the right side of the equation: log9 + (5 - 4log3 + x) = log3(3^2) + (5 - 4log3 + x) = 2log3 + (5 - 4log3 + x)

Step 3: Substitute the simplified expressions back into the equation: log3(x^2) = 2log3 + (5 - 4log3 + x)

Step 4: Simplify the equation further: log3(x^2) = 2 + 5 - 4log3 + x

Step 5: Combine like terms: log3(x^2) + 4log3 = 7 + x

Step 6: Combine the logarithms on the left side of the equation: log3(x^2) + log3(3^4) = 7 + x log3(x^2 * 3^4) = 7 + x

Step 7: Simplify the logarithm on the left side of the equation: log3(81x^2) = 7 + x

Step 8: Remove the logarithm by exponentiating both sides of the equation with base 3: 81x^2 = 3^(7 + x)

Step 9: Simplify the right side of the equation: 81x^2 = 3^7 * 3^x 81x^2 = 2187 * 3^x

Step 10: Divide both sides of the equation by 81: x^2 = (2187 * 3^x) / 81 x^2 = 27 * 3^x

Step 11: Rewrite 3^x as (3^1)^x: x^2 = 27 * (3^1)^x x^2 = 27 * 3^x

At this point, it is not possible to solve the equation algebraically. You can use numerical methods like graphing, iteration, or a calculator to approximate the value(s) of x that satisfy the equation.