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Solve for xx, rounding to the nearest hundredth. 91.2x=2791.2^x = 27

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Full solution

Q. Solve for xx, rounding to the nearest hundredth. 91.2x=2791.2^x = 27
  1. Given Equation: We are given the equation 91.2x=2791.2^x = 27 and we need to solve for xx. To do this, we will take the logarithm of both sides of the equation to isolate xx. We can use any logarithm base, but it's common to use base 1010 or the natural logarithm (ln⁡\ln). Let's use the natural logarithm for this calculation.
  2. Apply Natural Logarithm: Apply the natural logarithm to both sides of the equation:\newlineln⁡(91.2x)=ln⁡(27)\ln(91.2^x) = \ln(27)
  3. Simplify Left Side: Use the logarithmic property that ln⁡(ab)=b⋅ln⁡(a)\ln(a^b) = b\cdot\ln(a) to simplify the left side of the equation:\newlinex⋅ln⁡(91.2)=ln⁡(27)x \cdot \ln(91.2) = \ln(27)
  4. Divide to Solve for x: Divide both sides of the equation by ln⁡(91.2)\ln(91.2) to solve for x:\newlinex=ln⁡(27)ln⁡(91.2)x = \frac{\ln(27)}{\ln(91.2)}
  5. Calculate xx: Use a calculator to find the values of ln⁡(27)\ln(27) and ln⁡(91.2)\ln(91.2), and then divide them to find xx:
    x≈ln⁡(27)ln⁡(91.2)x \approx \frac{\ln(27)}{\ln(91.2)}
    x≈3.29584.5118x \approx \frac{3.2958}{4.5118}
    x≈0.7303x \approx 0.7303
  6. Round to Nearest Hundredth: Round the answer to the nearest hundredth as the question prompt asks:\newlinex≈0.73x \approx 0.73

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