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$2 = 1.08^n$

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Solve exponential equations by rewriting the base

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Q.

2 = 1.08^n

Apply Logarithm: We are given the equation $2 = 1.08^n$ and we need to find the value of $n$ . To solve for $n$ , we can take the logarithm of both sides of the equation. We can use any logarithm base, but it's common to use base $10$ or the natural logarithm base $e$ . Let's use the natural logarithm ( $\ln$ ) for this calculation. We apply the logarithm to both sides: $\ln(2) = \ln(1.08^n)$ .
Rewrite Equation: Using the property of logarithms that allows us to bring the exponent in front of the logarithm, we rewrite the right side of the equation: $\ln(2) = n \times \ln(1.08)$ .
Isolate n: Now we need to isolate n. To do this, we divide both sides of the equation by $\ln(1.08)$ : $n = \frac{\ln(2)}{\ln(1.08)}$ .
Calculate $n$ : We can now calculate the value of $n$ using a calculator: $n \approx \frac{\ln(2)}{\ln(1.08)}$ . Using a calculator, we find that $\ln(2) \approx 0.693147$ and $\ln(1.08) \approx 0.076961$ . So, $n \approx \frac{0.693147}{0.076961}$ .
Final Result: Performing the division, we get $n \approx 9.0055$ . This is the value of $n$ that satisfies the equation $2 = 1.08^n$ .

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