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Takumi plants a tree in his backyard and studies how the number of branches grows over time. He predicts that the relationship between N, the number of branches on the tree, and t, the elapsed time, in years, since the tree was planted can be modeled by the following equation. N=5*10^(0.3 t) According to Takumi's model, in how many years will the tree have 100 branches? Give an exact answer expressed as a base-10 logarithm. years

Takumi plants a tree in his backyard and studies how the number of branches grows over time.\newlineHe predicts that the relationship between N N , the number of branches on the tree, and t t , the elapsed time, in years, since the tree was planted can be modeled by the following equation.\newlineN=5⋅100.3t N=5 \cdot 10^{0.3 t} \newlineAccording to Takumi's model, in how many years will the tree have 100100 branches?\newlineGive an exact answer expressed as a base−10-10 logarithm.\newlineyears

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Q. Takumi plants a tree in his backyard and studies how the number of branches grows over time.\newlineHe predicts that the relationship between N N , the number of branches on the tree, and t t , the elapsed time, in years, since the tree was planted can be modeled by the following equation.\newlineN=5⋅100.3t N=5 \cdot 10^{0.3 t} \newlineAccording to Takumi's model, in how many years will the tree have 100100 branches?\newlineGive an exact answer expressed as a base−10-10 logarithm.\newlineyears
  1. Write Equation and Value: Write down the given equation and the value for NN that we want to achieve.\newlineWe are given the equation N=5×10(0.3t)N = 5 \times 10^{(0.3t)} and we want to find the value of tt when N=100N = 100.
  2. Set Up and Solve: Set up the equation with N=100N = 100 and solve for tt.100=5×10(0.3t)100 = 5 \times 10^{(0.3t)}
  3. Isolate Exponential Term: Divide both sides of the equation by 55 to isolate the exponential term.\newline100/5=100.3t100 / 5 = 10^{0.3t}\newline20=100.3t20 = 10^{0.3t}
  4. Apply Logarithm: Apply the logarithm to both sides of the equation to solve for tt.log⁡(20)=log⁡(100.3t)\log(20) = \log(10^{0.3t})
  5. Simplify Equation: Use the property of logarithms that log⁡(ab)=b⋅log⁡(a)\log(a^b) = b \cdot \log(a) to simplify the right side of the equation.log⁡(20)=0.3t⋅log⁡(10)\log(20) = 0.3t \cdot \log(10)
  6. Divide and Solve: Since log⁡(10)\log(10) is 11, we can simplify the equation further.\newlinelog⁡(20)=0.3t\log(20) = 0.3t
  7. Final Answer: Divide both sides of the equation by 0.30.3 to solve for tt. \newlinet=log⁡(20)0.3t = \frac{\log(20)}{0.3}
  8. Final Answer: Divide both sides of the equation by 0.30.3 to solve for tt.\newlinet=log⁡(20)0.3t = \frac{\log(20)}{0.3}Express the final answer as a base-1010 logarithm.\newlinet=log⁡(20)log⁡(100.3)t = \frac{\log(20)}{\log(10^{0.3})}

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