Model the pair of situations with exponential functions $$ f $$ and $$ g $$. Find the approximate value of $$ x $$ that makes $$ f(x) $$ $$ =g(x) $$. f. initial value of 700 decreasing at a rate of $$ 15 \% $$ $$ g $$ : initial value of 20 increasing at a rate of $$ 15 \% $$ $$ f(x)=700(0.85)^{x} $$ (Use integers or decimals for any numbers in the expression.) Write the exponential function for $$ g $$. $$ g(x)=\square $$ (Use integers or decimals for any numbers in the expression.)

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To model the pair of situations with exponential functions $$ f $$ and $$ g $$, we need to find the exponential function for $$ g $$ first.

Given that the initial value of $$ g $$ is 20 and it is increasing at a rate of 15%, the exponential function for $$ g $$ can be written as:

$$ g(x) = 20(1.15)^x $$

Now, we need to find the approximate value of $$ x $$ that makes $$ f(x) = g(x) $$.

To find the value of $$ x $$ where $$ f(x) = g(x) $$, we can set the two functions equal to each other and solve for $$ x $$:

$$ 700(0.85)^x = 20(1.15)^x $$

Let's solve this equation to find the approximate value of $$ x $$.
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 approximate value of $$ x $$ that makes $$ f(x) = g(x) $$ is $$ x \approx 11.761737 $$.
The exponential function for $$ g $$ is $$ g(x) = 20(1.15)^x $$. The approximate value of $$ x $$ where $$ f(x) = g(x) $$ is $$ x \approx 11.76 $$.

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