I have 2 problems I need help with both I have narrowed it down to 2 possible answers but can't decide which one is right. The first problem log subscript 5 (2x+3)=2 the choices a)13/2 and b) 14. The next problem 8^x = 16^x+2 the choices a)8 and b)-8. Help please!
Given that information I want to say the answer is -8 ?
To solve the first problem, you need to isolate the variable x. Let's go through it step by step:
1. Start by applying the logarithmic property that says log base a of b is equal to c is the same as saying a raised to the power of c is equal to b. In this case, log subscript 5 (2x+3)=2 translates to 5 raised to the power of 2 equals 2x+3.
So, 5^2 = 2x + 3. Simplifying, 25 = 2x + 3.
2. Next, isolate the variable by subtracting 3 from both sides: 25 - 3 = 2x.
Simplifying further, 22 = 2x.
3. Now, divide both sides by 2 to solve for x: 22/2 = x.
So, x = 11.
Now, let's move on to the second problem:
1. Start by applying the exponent property that states a^c * b^c is equal to (a * b)^c. In this case, 8^x = 16^x+2 can be rewritten as (2^3)^x = (2^4)^x+2.
So, 2^(3x) = 2^(4x + 8).
2. According to the property a^x = b^x, if the bases are equal, then the exponents must also be equal. Therefore, we can equate the exponents:
3x = 4x + 8.
3. Now, solve for x by subtracting 4x from both sides: 3x - 4x = 8.
Simplifying further, -x = 8.
4. Multiply both sides by -1 to solve for x: (-1)(-x) = (-1)(8).
Simplifying further, x = -8.
Based on the given information, the correct answers are:
1. Problem 1: x = 11 (Option b is correct).
2. Problem 2: x = -8 (Option b is correct).
Therefore, the answer to the second problem is -8, not -8.