Question

Найдите значение выражения: 2㏒²7 (49)=?

Answer 1

Найти значение выражения.

# 1 способ

$2log_7^2(49) = 2\left(log_7(7^2)\right)^2 = 2\left(2log_7(7)\right)^2 = 2*(2)^2 = 2 * 4 = 8.$

# 2 способ

$2log_7^2(49) = 2\left(log_7(49)\right)^2 = \left(\sqrt{2}log_7(49)\right)^2 = \left(2^{1/2}log_7(7^2)\right)^2 =\\\left(log_7((7^2)^{2^{1/2}})\right)^2 = \left(log_7(7^{2*2^{1/2}}})\right)^2 = \left(log_7(7^{2^{1+1/2}})\right)^2 =\\\left(log_7(7^{2^{3/2}})\right)^2 = \left(2^{3/2}log_7(7)\right)^2 = (2^{3/2})^2 = 2^{3/2*2} = 2^3 = 8.$

Ответ: 8.

Пример из комментариев.

# 1 способ

$\left(2*log_7(49)\right)^2 = \left(log_7((7^2)^2)\right)^2 = \left(log_7(7^{2*2})\right)^2 = \left(log_7(7^4)\right)^2 =\\= \left(4log_7(7)\right)^2 = 4^2 = 16.$

# 2 способ

$\left(2*log_7(49)\right)^2 = \left(2*log_7(7^2)\right)^2 = \left(2*2*log_7(7)\right)^2 = (4)^2 = 16.$

Ответ: 16.

Второй пример из комментариев.

$log^2_{\sqrt{7}}(49) = \left(log_{7^{\frac{1}{2}}}(7^{2})\right)^2 = \left(2*2*log_{7}(7)\right)^2 = 4^2 = 16.$

Ответ: 16.

Answer 2

To find the value of the expression 2㏒²₇ (49), we need to perform the following steps:

Step 1: Calculate the logarithm of 49 to the base 7. Step 2: Square the result. Step 3: Multiply the squared result by 2.

Step 1: Calculate the logarithm of 49 to the base 7. ㏒₇(49) = ?

The logarithm of 49 to the base 7 can be represented as follows: ㏒₇(49) = x ⟺ 7^x = 49

To find the value of x, we need to determine what power (x) we raise 7 to in order to get 49. It turns out that 7^2 = 49, so x = 2.

Step 2: Square the result. (㏒₇(49))^2 = 2^2 = 4

Step 3: Multiply the squared result by 2. 2 × 4 = 8

So, the value of the expression 2㏒²₇ (49) is 8.