# Exponential Equations solve 27^x=1/√3

• MHB
• Fright535
In summary, to solve the equation 27^x = 1/√3, we can use logarithms by rewriting 27^x as 3^(3x) and taking the logarithm of both sides. It is also possible to solve without logarithms by rewriting 1/√3 as √3/3 and taking the cube root. The exact solution to this equation is x = log(√3) / log(3) or 1/2. The equation only has one solution and it can be checked by plugging the solution back into the original equation.
Fright535
I'm taking an online class and I was doing some very simple exponential equations when this was thrown at me, and I have no clue how to solve it.

27^x=1/√3

Hello, and welcome to MHB! (Wave)

We are given to solve:

$$\displaystyle 27^x=\frac{1}{\sqrt{3}}$$

Can you write both sides of the equation as a power of 3?

To follow up, we may write:

$$\displaystyle 3^{3x}=3^{-\frac{1}{2}}$$

Since we have like bases, we can simply equate the exponents:

$$\displaystyle 3x=-\frac{1}{2}\implies x=-\frac{1}{6}$$

## 1. How do I solve 27^x = 1/√3?

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.

## 2. Can this equation be solved without using logarithms?

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.

## 3. What is the exact solution to this equation?

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.

## 4. Can this equation have multiple solutions?

No, this equation only has one solution. When solving exponential equations, we can only have one unique solution.

## 5. How can I check if my solution is correct?

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.

• General Math
Replies
1
Views
747
• General Math
Replies
3
Views
1K
• General Math
Replies
2
Views
1K
• General Math
Replies
3
Views
1K
• General Math
Replies
9
Views
412
• Precalculus Mathematics Homework Help
Replies
12
Views
1K
• Differential Equations
Replies
7
Views
2K
• General Math
Replies
13
Views
1K
• General Math
Replies
2
Views
1K
• General Math
Replies
7
Views
1K