I have 3 problems I need my answers confirmed for. The solution for log subscript 3 1/9 = x is the solution -1/2? Secondl, what is the solution for 9^2x = 3^x+6 does x=2? Lastley is the answer to 3 ^5x+2 = 81 does x= 2/5? Help sincerely appreciated!
The solution for log subscript 3 1/9 = x is the solution -1/2?
No.
Secondl, what is the solution for 9^2x = 3^x+6 does x=2?
9^2x = 3^4x So no
Lastley is the answer to 3 ^5x+2 = 81 does x= 2/5?
Yes
To confirm the first problem, we need to solve the equation log₃(1/9) = x.
To do this, we need to understand that logarithms are the inverse function of exponentiation. In this case, we are looking for the exponent that would give us 1/9 when we raise 3 to that power.
Rewriting the equation, we have 3^x = 1/9.
To solve for x, we can express 1/9 as a power of 3. We know that 1/9 is equal to 3^(-2) because raising 3 to the power of -2 yields 1/3^2 = 1/9.
Now we have 3^x = 3^(-2). Since the bases (3) are the same, we can set the exponents equal to each other.
Therefore, x = -2.
So, the solution for log₃(1/9) = x is x = -2.
For the second problem, we have the equation 9^(2x) = 3^(x+6).
To solve this equation, we need to rewrite both sides using the same base. Since 9 (3^2) and 3 have different bases, we can rewrite 9 as 3^2.
The equation becomes (3^2)^(2x) = 3^(x+6). Applying the power rule of exponents, we can simplify the left side to 3^(4x).
So, we have 3^(4x) = 3^(x+6).
Since the bases are the same, we can set the exponents equal to each other.
This gives us the equation 4x = x + 6.
Simplifying further, we have 3x = 6.
Dividing both sides by 3, we find that x = 2.
However, this means that x = 2 is NOT the solution to the equation 9^(2x) = 3^(x+6).
Finally, for the third problem, the equation is 3^(5x+2) = 81.
We can rewrite 81 as 3^4, since 81 = 3^4.
Thus, we have 3^(5x+2) = 3^4.
Since the bases are the same, we set the exponents equal to each other.
This gives us 5x + 2 = 4.
Solving for x, we subtract 2 from both sides:
5x = 2.
Dividing both sides by 5, we find that x = 2/5.
Therefore, x = 2/5 is the solution to the equation 3^(5x+2) = 81.
I hope this explanation helps!