Question

X^㏒₃X=81 ............................

Answer 1

x > 0

$x^{log_{3}x }=81\\\\log_{3}x^{log_{3}x }=log_{3}81\\\\log_{3} x*log_{3}x=log_{3}3^{4}\\\\log^{2}_{3}x=4\\\\1)log_{3}x=2\\\\x_{1}=3^{2}=9\\\\2)log_{3}x=-2\\\\x_{2}=3^{-2}=\frac{1}{9}\\\\OTVET:\frac{1}{9};9$

Answer 2

Ответ:

Объяснение:

$x^{log_3(x)}=81;x>0\\log_x(x^{log_3(x)})=log_x(81)\\log_3(x)=log_x(81)\\log_3(x)=\frac{log_3(81)}{log_3(x)} \\log_3^2(x)=4\\log_3(x)=-2=>x=\frac{1}{9}\\log_3(x)=2=>x=9$

Answer 3

The equation you've provided is $x^{\log_3 x} = 81$. To solve this equation, you can use logarithms to isolate the variable $x$. Here's how you can do it:

1. Start by taking the natural logarithm (ln) of both sides of the equation to remove the exponent:

   $\ln(x^{\log_3 x}) = \ln(81)$

2. Use the property of logarithms that allows you to bring the exponent down as a coefficient:

   $\log_3(x) \cdot \ln(x) = \ln(81)$

3. Now, we can use the change of base formula to convert the base of the first logarithm to a common base (e.g., natural logarithm or base 10):

   $\frac{\ln(x)}{\ln(3)} \cdot \ln(x) = \ln(81)$

4. Multiply both sides by $\ln(3)$ to simplify:

   $\ln(x) \cdot \ln(x) = \ln(3) \cdot \ln(81)$

5. Take the square root of both sides to isolate $\ln(x)$:

   $\ln(x) = \sqrt{\ln(3) \cdot \ln(81)}$

6. Now, exponentiate both sides to get rid of the natural logarithm:

   $x = e^{\sqrt{\ln(3) \cdot \ln(81)}}$

You can then calculate the value of $x$ using a calculator. This will give you the approximate solution to the equation $x^{\log_3 x} = 81$.