(1/2)(2/3)^x=(1/4)(16/27)

Suzhoumeite

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Jun 12, 2024

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Solving the Equation: (1/2)(2/3)^x = (1/4)(16/27)

In this article, we will solve the equation:

$(1/2)(2/3)^x = (1/4)(16/27)$

Step 1: Simplify the Right-Hand Side

Let's start by simplifying the right-hand side of the equation:

$\frac{1}{4} \cdot \frac{16}{27} = \frac{4}{27}$

So, the equation becomes:

$(1/2)(2/3)^x = \frac{4}{27}$

Step 2: Simplify the Left-Hand Side

Now, let's simplify the left-hand side of the equation:

$(1/2)(2/3)^x = \frac{1}{2} \cdot \left(\frac{2}{3}\right)^x$

Step 3: Equate and Solve

Now, we can equate the two expressions:

$\frac{1}{2} \cdot \left(\frac{2}{3}\right)^x = \frac{4}{27}$

To solve for x, we can start by multiplying both sides by 2:

$\left(\frac{2}{3}\right)^x = \frac{8}{27}$

Next, we can take the logarithm of both sides:

$x \log\left(\frac{2}{3}\right) = \log\left(\frac{8}{27}\right)$

Now, we can solve for x:

$x = \frac{\log\left(\frac{8}{27}\right)}{\log\left(\frac{2}{3}\right)}$

Solution

After simplifying the expression, we get:

$x = 3$

Therefore, the solution to the equation is x = 3.

I hope this helps! Let me know if you have any questions.