Is the solution for log sub 3 1/9 = x does x = -2? Is the answer for 9^2x = 3^x+6 is it 0?
Yes, No
To find the solution for logarithmic equations, we can use the properties of logarithms and algebraic manipulation.
1. For the equation `log(sub3)1/9 = x`, we can rewrite it in exponential form as `3^x = 1/9`. Since `1/9` can be expressed as `(1/3)^2`, the equation becomes `3^x = (1/3)^2`. To solve for x, we can equate the bases: `3^x = 3^(-2)`. Since the bases are the same, the exponents must be equal. Therefore, x equals -2.
2. For the equation `9^2x = 3^(x+6)`, we can rewrite both sides with the same base (3 or 9) and then set the exponents equal to each other.
Starting with the left side, we rewrite `9` as `3^2`. The equation becomes `(3^2)^(2x) = 3^(x+6)`. Applying the exponent multiplication rule, we have `3^(4x) = 3^(x+6)`.
Since the bases are the same, we can equate the exponents: `4x = x+6`. We solve this linear equation for x:
4x - x = 6
3x = 6
x = 2
Therefore, the solution for `9^2x = 3^(x+6)` is x = 2.
In summary:
- The equation `log(sub3)1/9 = x` gives x = -2 as the solution.
- The equation `9^2x = 3^(x+6)` gives x = 2 as the solution.