Given Equation: We are given the equation 0.25 = 0.95^x and we need to solve for x. To solve for x, we can take the logarithm of both sides of the equation. Let's use the natural logarithm (ln) for this purpose.Take Natural Logarithm: Take the natural logarithm of both sides of the equation: ln(0.25) = ln(0.95^x).Rewrite Using Property: Using the property of logarithms that ln(a^b) = b * ln(a), we can rewrite the right side of the equation: ln(0.25) = x * ln(0.95).Calculate Natural Logarithms: Now we need to calculate the natural logarithms of 0.25 and 0.95. ln(0.25) ≈ -1.3863 ln(0.95) ≈ -0.0513Substitute Values: Substitute the values of the natural logarithms back into the equation: -1.3863 = x * (-0.0513).Solve for x: To solve for x, divide both sides of the equation by -0.0513: x = -1.3863 / -0.0513.Perform Division: Perform the division to find the value of x: x ≈ 27.027

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$0.25=0.95^x$

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Precalculus

Solve exponential equations by rewriting the base

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0.25=0.95^x

Given Equation: We are given the equation $0.25 = 0.95^x$ and we need to solve for $x$ . $\newline$ To solve for $x$ , we can take the logarithm of both sides of the equation. $\newline$ Let's use the natural logarithm $(\ln)$ for this purpose.
Take Natural Logarithm: Take the natural logarithm of both sides of the equation: $\newline$ $\ln(0.25) = \ln(0.95^x)$ .
Rewrite Using Property: Using the property of logarithms that $\ln(a^b) = b \cdot \ln(a)$ , we can rewrite the right side of the equation: $\ln(0.25) = x \cdot \ln(0.95)$ .
Calculate Natural Logarithms: Now we need to calculate the natural logarithms of $0.25$ and $0.95$ . $\newline$ $\ln(0.25) \approx -1.3863$ $\newline$ $\ln(0.95) \approx -0.0513$
Substitute Values: Substitute the values of the natural logarithms back into the equation: $\newline$ $-1.3863 = x \times (-0.0513)$ .
Solve for x: To solve for x, divide both sides of the equation by $-0.0513$ : $\newline$ $x = \frac{-1.3863}{-0.0513}.$
Perform Division: Perform the division to find the value of $x$ : $x \approx 27.027$