He predicts that the relationship between N, the number of branches on the tree, and t years, since the tree was planted can be modeled by the following equation. N = 5. 100.3t 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.

Set Up Equation: We start by setting up the equation given by Takumi's model where N, the number of branches, equals 100. N = 5 * 100.3^t 100 = 5 * 100.3^tIsolate Exponential Term: Next, we divide both sides by 5 to isolate the exponential term. 100 / 5 = 100.3^t 20 = 100.3^tApply Logarithm: Now, we apply the logarithm to both sides to solve for t. We use the base-10 logarithm. log_10(20) = log_10(100.3^t)Bring Down Exponent: Using the power property of logarithms, we can bring down the exponent. log_10(20) = t * log_10(100.3)Solve for t: Finally, solve for t by dividing both sides by log_10(100.3). t = log_10(20) / log_10(100.3)

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He predicts that the relationship between $N$ , the number of branches on the tree, and $t$ years, since the tree was planted can be modeled by the following equation. $N = 5 \cdot 100.3^t$ 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.

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Algebra 2

Properties of logarithms: mixed review

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Q. He predicts that the relationship between

N

, the number of branches on the tree, and

t

years, since the tree was planted can be modeled by the following equation.

N = 5 \cdot 100.3^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.

Set Up Equation: We start by setting up the equation given by Takumi's model where $N$ , the number of branches, equals $100$ .
$N = 5 \times 100.3^t$
$100 = 5 \times 100.3^t$
Isolate Exponential Term: Next, we divide both sides by $5$ to isolate the exponential term. $\newline$ $\frac{100}{5} = 100.3^t$ $\newline$ $20 = 100.3^t$
Apply Logarithm: Now, we apply the logarithm to both sides to solve for $t$ . We use the base- $10$ logarithm. $\newline$ $\log_{10}(20) = \log_{10}(100.3^t)$
Bring Down Exponent: Using the power property of logarithms, we can bring down the exponent. $\log_{10}(20) = t \cdot \log_{10}(100.3)$
Solve for t: Finally, solve for $t$ by dividing both sides by $\log_{10}(100.3)$ .
$t = \frac{\log_{10}(20)}{\log_{10}(100.3)}$