Questions: Use algebraic procedures to find the exact solution [ left(frac52right)^x=frac1258 ]

$Use algebraic procedures to find the exact solution [ left(frac52right)^x=frac1258 ]$

Transcript text: Use algebraic procedures to find the exact solution \[ \left(\frac{5}{2}\right)^{x}=\frac{125}{8} \]

Solution

Solution Steps

To solve the equation $\left(\frac{5}{2}\right)^{x}=\frac{125}{8}$, we can use logarithms to isolate $x$. By taking the natural logarithm (or any logarithm) on both sides, we can then solve for $x$.

Step 1: Rewrite the Equation

We start with the equation: \[ \left(\frac{5}{2}\right)^{x} = \frac{125}{8} \]

Step 2: Express the Right Side in Terms of the Base

We can express $\frac{125}{8}$ as: \[ \frac{125}{8} = \frac{5^3}{2^3} = \left(\frac{5}{2}\right)^{3} \]

Step 3: Set the Exponents Equal

Since the bases are the same, we can set the exponents equal to each other: \[ x = 3 \]

Final Answer

Thus, the solution to the equation is: \[ \boxed{x = 3} \]

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Questions: Use algebraic procedures to find the exact solution [ left(frac52right)^x=frac1258 ]

Solution

Solution Steps

Final Answer

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