Solve the equation: 4^(2x+1) * 5^(x-2) = 6^(1-x)

Steps to Solve

1. Apply the logarithm to both sides of the equation We can apply logarithms to both sides of the equation, using the property that if $a = b$, then $\log(a) = \log(b)$. Using the natural logarithm (ln) helps simplify the equation.

   $ln(4^{2x+1} \times 5^{x-2}) = ln(6^{1-x})$

2. Use logarithm properties to expand the equation Apply the logarithm properties: $\log(ab) = \log(a) + \log(b)$ and $\log(a^b) = b\log(a)$:

   $ln(4^{2x+1}) + ln(5^{x-2}) = ln(6^{1-x})$ $(2x+1)ln(4) + (x-2)ln(5) = (1-x)ln(6)$

3. Expand the terms Distribute the $ln$ terms across the parentheses:

   $2x \cdot ln(4) + ln(4) + x \cdot ln(5) - 2 \cdot ln(5) = ln(6) - x \cdot ln(6)$

4. Collect terms with 'x' on one side and constants on the other side Rearrange the equation to group the terms containing $x$ on the left side and the constant terms on the right side:

   $2x \cdot ln(4) + x \cdot ln(5) + x \cdot ln(6) = ln(6) - ln(4) + 2 \cdot ln(5)$

5. Factor out 'x' from the left side Factor out $x$ from the left side of the equation:

   $x(2 \cdot ln(4) + ln(5) + ln(6)) = ln(6) - ln(4) + 2 \cdot ln(5)$

6. Isolate 'x' Divide both sides by $(2 \cdot ln(4) + ln(5) + ln(6))$ to solve for $x$:

   $x = \frac{ln(6) - ln(4) + 2 \cdot ln(5)}{2 \cdot ln(4) + ln(5) + ln(6)}$

7. Simplify using logarithm properties Use properties of logarithms to simplify: $a \cdot ln(b) = ln(b^a)$, $ln(a) + ln(b) = ln(ab)$, and $ln(a) - ln(b) = ln(\frac{a}{b})$.

   $x = \frac{ln(6) - ln(4) + ln(5^2)}{ln(4^2) + ln(5) + ln(6)}$

   $x = \frac{ln(6) - ln(4) + ln(25)}{ln(16) + ln(5) + ln(6)}$

   $x = \frac{ln(\frac{6}{4}) + ln(25)}{ln(16) + ln(30)}$

   $x = \frac{ln(\frac{3}{2}) + ln(25)}{ln(480)}$

   $x = \frac{ln(\frac{3}{2} \times 25)}{ln(480)}$

   $x = \frac{ln(\frac{75}{2})}{ln(480)}$

   $x = \frac{ln(37.5)}{ln(480)}$

8. Final Answer

   $x = \frac{ln(37.5)}{ln(480)} \approx \frac{3.6243}{6.1738} \approx 0.5870$