Take the natural logarithm on both sides of the equation.

Mathematics

ScanSolve AI Answer

Answer

2.07

Step-by-step solution

4 steps

  1. Take the natural logarithm on both sides of the equation.

    3x+4=52x3^{x+4} = 5^{2x} ln⁡(3x+4)=ln⁡(52x)\ln(3^{x+4}) = \ln(5^{2x})

  2. Use the logarithm property ln⁡(ab)=bln⁡(a)\ln(a^b) = b \ln(a) to bring down the exponents.

    (x+4)ln⁡(3)=2xln⁡(5)(x+4)\ln(3) = 2x\ln(5)

  3. Expand and rearrange the equation to solve for xx.

    xln⁡(3)+4ln⁡(3)=2xln⁡(5)x\ln(3) + 4\ln(3) = 2x\ln(5) 4ln⁡(3)=2xln⁡(5)−xln⁡(3)4\ln(3) = 2x\ln(5) - x\ln(3) 4ln⁡(3)=x(2ln⁡(5)−ln⁡(3))4\ln(3) = x(2\ln(5) - \ln(3)) x=4ln⁡(3)2ln⁡(5)−ln⁡(3)x = \frac{4\ln(3)}{2\ln(5) - \ln(3)}

  4. Calculate the numerical value of xx and round to 3 significant figures.

    x=4×1.09862×1.6094−1.0986x = \frac{4 \times 1.0986}{2 \times 1.6094 - 1.0986} x=4.39443.2188−1.0986x = \frac{4.3944}{3.2188 - 1.0986} x=4.39442.1202x = \frac{4.3944}{2.1202} x≈2.07263465x \approx 2.07263465

    Rounding to 3 significant figures:

    x≈2.07x \approx 2.07

    The final answer is 2.07.

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