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.