Model the pair of situations with exponential functions \( f \) and \( g \). Find the approximate value of \( x \) tha \( =g(x) \). f . initial value of 700 decreasing at a rate of \( 15 \% \) \( g \) : initial value of 20 increasing at a rate of \( 15 \% \) wrme me exponenual runction ror \( g \). \( g(x)=20(1.15)^{x} \) (Use integers or decimals for any numbers in the expression.) The value of \( x \) that makes \( f(x)=g(x) \) is \( x=\square \). (Round to the nearest integer as needed.)

To model the pair of situations with exponential functions \( f \) and \( g \), we need to find the exponential function for \( f \) and then solve for the value of \( x \) where \( f(x) = g(x) \). Given: - \( f \) has an initial value of 700 and decreases at a rate of 15%. - \( g \) has an initial value of 20 and increases at a rate of 15%. The exponential function for \( g \) is given as \( g(x) = 20(1.15)^x \). To find the exponential function for \( f \), we can use the formula for exponential decay: \[ f(x) = a(1 - r)^x \] where: - \( a \) is the initial value, - \( r \) is the rate of decrease. For \( f \), the initial value is 700 and the rate of decrease is 15%, so: \[ f(x) = 700(1 - 0.15)^x \] \[ f(x) = 700(0.85)^x \] Now, we need to solve for \( x \) where \( f(x) = g(x) \): \[ 700(0.85)^x = 20(1.15)^x \] To solve this equation, we can use the logarithmic method. Taking the logarithm of both sides will allow us to simplify the equation and solve for \( x \). Solve the equation by following steps: - step0: Solve for \(x\): \(700\times 0.85^{x}=20\times 1.15^{x}\) - step1: Convert the expressions: \(700\left(\frac{17}{20}\right)^{x}=20\left(\frac{23}{20}\right)^{x}\) - step2: Multiply both sides: \(\frac{1}{700}\times 700\left(\frac{17}{20}\right)^{x}=\frac{1}{700}\times 20\left(\frac{23}{20}\right)^{x}\) - step3: Calculate: \(\left(\frac{17}{20}\right)^{x}=\frac{1}{35}\left(\frac{23}{20}\right)^{x}\) - step4: Divide both sides: \(\frac{\left(\frac{17}{20}\right)^{x}}{\left(\frac{23}{20}\right)^{x}}=\frac{1}{35}\) - step5: Evaluate: \(\left(\frac{17}{23}\right)^{x}=\frac{1}{35}\) - step6: Take the logarithm of both sides: \(\log_{\frac{17}{23}}{\left(\left(\frac{17}{23}\right)^{x}\right)}=\log_{\frac{17}{23}}{\left(\frac{1}{35}\right)}\) - step7: Evaluate the logarithm: \(x=\log_{\frac{17}{23}}{\left(\frac{1}{35}\right)}\) - step8: Simplify: \(x=-\log_{\frac{17}{23}}{\left(35\right)}\) The value of \( x \) that makes \( f(x) = g(x) \) is approximately \( x \approx 11.761737 \). Rounding to the nearest integer, we get \( x \approx 12 \).