Question

A) 4^ 3x+5 =4^ 3-5x. ; b) 7 ^ (4x + 5) = 7 ^ (9 - 5x) ;c) 6 ^ (x + 5) = 6 ^ (3x) ;d) 8^ x+5 =8^ 2-5x ;e) 11 ^ x = 11 ^ (2 + 5x) ;f) 2 ^ (x - 5) = 2 ^ (25x). ​

Answer 1

Ответ:

$a) \: x = - 0.25 \\ b) \: \: x = \frac{4}{9} \: \: \: \: \: \: \: \: \\ c) \: \: x = 2.5 \: \: \: \: \: \: \\d) \: \: x = - 0.5 \: \: \\e) \: \: x = - 0.5 \: \: \\ f) \: \: x = - \frac{5}{24} \: \: \:$

Решение на фото

Answer 2

A) To solve the equation 4^3x + 5 = 4^3 - 5x, we can start by simplifying both sides of the equation.

On the left side, we have 4^3x + 5. We can rewrite 4^3 as 64, so the equation becomes 64x + 5.

On the right side, we have 4^3 - 5x. Again, 4^3 is 64, so the equation becomes 64 - 5x.

Now we have the equation 64x + 5 = 64 - 5x.

To solve for x, we can combine like terms by adding 5x to both sides of the equation. This gives us 64x + 5x + 5 = 64 - 5x + 5x.

Simplifying further, we have 69x + 5 = 64.

Next, we can isolate the variable x by subtracting 5 from both sides of the equation. This gives us 69x = 59.

Finally, to solve for x, we divide both sides of the equation by 69. This gives us x = 59/69.

Therefore, the solution to the equation 4^3x + 5 = 4^3 - 5x is x = 59/69.

B) To solve the equation 7^(4x + 5) = 7^(9 - 5x), we can start by using the property of exponents which states that if two exponential expressions with the same base are equal, then their exponents must be equal.

Therefore, we can set the exponents equal to each other:

4x + 5 = 9 - 5x

To solve for x, we can first isolate the variable terms by adding 5x to both sides of the equation:

4x + 5x + 5 = 9

Combining like terms, we have:

9x + 5 = 9

Next, we can isolate the constant term by subtracting 5 from both sides of the equation:

9x = 9 - 5

Simplifying, we get:

9x = 4

Finally, to solve for x, we divide both sides of the equation by 9:

x = 4/9

Therefore, the solution to the equation 7^(4x + 5) = 7^(9 - 5x) is x = 4/9.

C) To solve the equation 6^(x + 5) = 6^(3x), we can again set the exponents equal to each other:

x + 5 = 3x

To isolate the variable terms, we can subtract x from both sides of the equation:

x + 5 - x = 3x - x

Simplifying, we have:

5 = 2x

Next, we can divide both sides of the equation by 2:

5/2 = x

Therefore, the solution to the equation 6^(x + 5) = 6^(3x) is x = 5/2.

D) To solve the equation 8^x + 5 = 8^2 - 5x, we can start by simplifying both sides of the equation:

On the left side, we have 8^x + 5.

On the right side, we have 8^2, which is 64, minus 5x.

Now we have the equation 8^x + 5 = 64 - 5x.

To solve for x, we can combine like terms by adding 5x to both sides of the equation. This gives us 8^x + 5x + 5 = 64.

Next, we can isolate the variable terms by subtracting 5 from both sides of the equation. This gives us 8^x + 5x = 59.

Unfortunately, we cannot solve this equation algebraically as it involves a variable exponent. To find the value of x, we would need to use numerical methods such as graphing or approximation techniques.

E) To solve the equation 11^x = 11^(2 + 5x), we can again set the exponents equal to each other:

x = 2 + 5x

Next, we can isolate the variable terms by subtracting 5x from both sides of the equation:

x - 5x = 2

Simplifying, we have:

-4x = 2

Finally, to solve for x, we divide both sides of the equation by -4:

x = 2/-4

Simplifying further, we get:

x = -1/2

Therefore, the solution to the equation 11^x = 11^(2 + 5x) is x = -1/2.

F) To solve the equation 2^(x - 5) = 2^(25x), we can again set the exponents equal to each other:

x - 5 = 25x

Next, we can isolate the variable terms by subtracting 25x from both sides of the equation:

x - 25x = 0

Simplifying, we have:

-24x = 0

Finally, to solve for x, we divide both sides of the equation by -24:

x = 0/-24

Simplifying further, we get:

x = 0

Therefore, the solution to the equation 2^(x - 5) = 2^(25x) is x = 0.

In summary, the solutions to the given equations are: A) x = 59/69 B) x = 4/9 C) x = 5/2 D) Numerical methods needed for finding x E) x = -1/2 F) x = 0