To solve this exponential equation, we can use logarithms. First, we can rewrite 27^x as (3^3)^x, which equals 3^(3x). Then, we can take the logarithm of both sides, giving us log(3^(3x)) = log(1/√3). Using the power rule of logarithms, we can bring down the exponent, giving us 3x * log(3) = log(1/√3). We can simplify the right side to be log(1) - log(√3), which equals 0 - log(√3). Therefore, we have 3x * log(3) = -log(√3). We can then solve for x by dividing both sides by 3 * log(3) and simplifying.
Yes, it is possible to solve this equation without using logarithms. We can rewrite 27^x as (3^3)^x, which equals 3^(3x). Then, we can rewrite 1/√3 as √3/3. This gives us the equation 3^(3x) = √3/3. We can then take the cube root of both sides, giving us 3x = (√3/3)^(1/3). We can simplify the right side to be (√3)^(1/3) / (3)^(1/3), which equals √3/∛27. Therefore, we have 3x = √3/∛27. We can then solve for x by dividing both sides by 3 and simplifying.
The exact solution to this equation is x = log(√3) / log(3), which can be simplified to x = 1/2. This means that when 27^x is equal to 1/√3, x must be equal to 1/2.
No, this equation only has one solution. When solving exponential equations, we can only have one unique solution.
You can check if your solution is correct by plugging it back into the original equation and solving. If the left side of the equation equals the right side, then your solution is correct. In this case, when x = 1/2, we have 27^(1/2) = √27 = 3, and 1/√3 = 1/(√3) = √3/3. Therefore, the left side does equal the right side, confirming that our solution is correct.