Question

Log3 √7*(8x+1) = 6 Решить логарифмическое уравнение

Answer 1

$log_{3}( \sqrt{7(8x + 1} ) = 6 \\ \frac{1}{2} \times log_{3}(7(8x + 1)) = 6 \\ log_{3}(7) + log_{3}(8x + 1) = 12 \\ log_{3}(8x + 1) = 12 - log_{3}(7) \\ 8x + 1 = \frac{{3}^{12} }{7 } \\ \ x = \frac{(\frac{{3}^{12} }{7 } - 1 )}{8} \\ x = 66430$

Answer 2

To solve the logarithmic equation log₃(√7*(8x+1)) = 6, we'll use the properties of logarithms. The main property we'll use is:

logₐ(b) = c is equivalent to a^c = b

Let's proceed with the solution:

Step 1: Convert the equation to exponential form: log₃(√7*(8x+1)) = 6 can be rewritten as 3^6 = √7*(8x+1)

Step 2: Solve for (8x+1): 3^6 = √7*(8x+1)

Since 3^6 = 729, the equation becomes: 729 = √7*(8x+1)

Step 3: Isolate (8x+1) on one side of the equation: Divide both sides by √7 to solve for (8x+1):

(8x+1) = 729/√7

Step 4: Calculate the value on the right-hand side: To simplify 729/√7, we can multiply both the numerator and denominator by √7:

(8x+1) = (729/√7) * (√7/√7) (8x+1) = (729*√7)/(√7 * √7)

Since √7 * √7 = 7, the equation becomes: (8x+1) = (729 * √7) / 7

Step 5: Calculate the value on the right-hand side: (8x+1) = (729 * √7) / 7 (8x+1) = 1041√7 / 7

Step 6: Isolate x on one side of the equation: Subtract 1 from both sides to solve for x:

8x = (1041√7 / 7) - 1

Step 7: Calculate the value on the right-hand side: 8x = (1041√7 / 7) - 1

Now, divide both sides by 8:

x = [(1041√7 / 7) - 1] / 8

This is the final solution for the logarithmic equation. However, if you need a numerical approximation for x, you can calculate the value using a calculator.