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Find the value of x that solves the equation ln(x-1)+3ln 2=ln 10. Answer: x=

Find the value of x x that solves the equation ln⁡(x−1)+3ln⁡2=ln⁡10 \ln (x-1)+3 \ln 2=\ln 10 .\newlineAnswer: x= x=

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Q. Find the value of x x that solves the equation ln⁡(x−1)+3ln⁡2=ln⁡10 \ln (x-1)+3 \ln 2=\ln 10 .\newlineAnswer: x= x=
  1. Simplify using logarithmic properties: First, we need to simplify the equation using logarithmic properties. The property that we can use here is that ln⁡(ab)=b⋅ln⁡(a)\ln(a^b) = b\cdot\ln(a), which means we can rewrite 3ln⁡(2)3\ln(2) as ln⁡(23)\ln(2^3).\newlineln⁡(x−1)+ln⁡(23)=ln⁡(10)\ln(x-1) + \ln(2^3) = \ln(10)
  2. Combine logarithms: Now, we can use another logarithmic property that states ln⁡(a)+ln⁡(b)=ln⁡(a∗b)\ln(a) + \ln(b) = \ln(a*b). We apply this property to combine the two logarithms on the left side of the equation.ln⁡((x−1)∗23)=ln⁡10\ln((x-1)*2^3) = \ln10
  3. Equate arguments: Since ln⁡(a)=ln⁡(b)\ln(a) = \ln(b) implies that a=ba = b, we can now equate the arguments of the logarithms.(x−1)⋅23=10(x-1)\cdot 2^3 = 10
  4. Calculate and multiply: Next, we calculate 232^3, which is 88, and then multiply it by (x−1)(x-1). \newline8(x−1)=108(x-1) = 10
  5. Distribute 88: Now, we distribute the 88 across the (x−1)(x-1).8x−8=108x - 8 = 10
  6. Isolate x: To isolate x, we add 88 to both sides of the equation.\newline8x−8+8=10+88x - 8 + 8 = 10 + 8\newline8x=188x = 18
  7. Add 88 to both sides: Finally, we divide both sides by 88 to solve for xx.x=188x = \frac{18}{8}x=2.25x = 2.25

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