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Find the value of 
x that solves the equation 
ln(x-3)-3ln 4=ln 10.
Answer: 
x=

Find the value of x x that solves the equation ln(x3)3ln4=ln10 \ln (x-3)-3 \ln 4=\ln 10 .\newlineAnswer: x= x=

Full solution

Q. Find the value of x x that solves the equation ln(x3)3ln4=ln10 \ln (x-3)-3 \ln 4=\ln 10 .\newlineAnswer: x= x=
  1. Write Equation: Write down the given equation.\newlineWe have the equation ln(x3)3ln4=ln10\ln(x-3) - 3\ln 4 = \ln 10.
  2. Simplify Logarithms: Use the property of logarithms that ln(ab)=bln(a)\ln(a^b) = b\cdot\ln(a) to simplify the term with ln4\ln 4.ln(x3)ln(43)=ln10\ln(x-3) - \ln(4^3) = \ln 10
  3. Recognize Value: Recognize that 434^3 is 6464. \newlineln(x3)ln(64)=ln10\ln(x-3) - \ln(64) = \ln 10
  4. Combine Logarithms: Use the property of logarithms that ln(a)ln(b)=ln(ab)\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) to combine the logarithms on the left side.\newlineln(x364)=ln10\ln\left(\frac{x-3}{64}\right) = \ln 10
  5. Set Equal: If ln(a)=ln(b)\ln(a) = \ln(b), then a=ba = b. So, set the inside of the logarithms equal to each other.x364=10\frac{x-3}{64} = 10
  6. Multiply by 6464: Solve for xx by multiplying both sides of the equation by 6464.x3=10×64x - 3 = 10 \times 64
  7. Calculate Result: Calculate 10×6410 \times 64. \newlinex3=640x - 3 = 640
  8. Isolate x: Add 33 to both sides to isolate xx.\newlinex=640+3x = 640 + 3
  9. Add 33: Calculate 640+3640 + 3.\newlinex=643x = 643

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