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 * 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 = ln(2) / ln(1.08).Calculate n: We can now calculate the value of n using a calculator: n ≈ ln(2) / ln(1.08). Using a calculator, we find that ln(2) ≈ 0.693147 and ln(1.08) ≈ 0.076961. So, n ≈ 0.693147 / 0.076961.Final Result: Performing the division, we get n ≈ 9.0055. This is the value of n that satisfies the equation 2 = 1.08^n.

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Precalculus

Solve exponential equations by rewriting the base

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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 $2 = 1.08^n$ .