Question

1. the total number of views, in thousands, of an online video is modeled by the function ( v(t) = 20log_{5}(t + 2) + 5 ), where ( t ) is the number of days since the video was posted online. describe the function as a transformation of its parent function, then graph the function. use the graph to determine how long it will take until the video has been viewed 50,000 times.

Explanation:

Step1: Set up the equation

We know that \( v(t) \) represents the number of views in thousands. We want to find \( t \) when the video has been viewed 50,000 times, so \( v(t)=50 \) (since 50,000 views is 50 thousand views). The function is \( v(t) = 20\log_{5}(t + 2)+5 \). So we set up the equation:
\[
50=20\log_{5}(t + 2)+5
\]

Step2: Isolate the logarithmic term

Subtract 5 from both sides of the equation:
\[
50 - 5=20\log_{5}(t + 2)
\]
\[
45 = 20\log_{5}(t + 2)
\]
Then divide both sides by 20:
\[
\log_{5}(t + 2)=\frac{45}{20}=\frac{9}{4}
\]

Step3: Convert from logarithmic to exponential form

Recall that if \( \log_{a}(b)=c \), then \( b = a^{c} \). So for \( \log_{5}(t + 2)=\frac{9}{4} \), we have:
\[
t + 2=5^{\frac{9}{4}}
\]

Step4: Calculate \( 5^{\frac{9}{4}} \) and solve for \( t \)

First, \( 5^{\frac{9}{4}}=\sqrt[4]{5^{9}}=\sqrt[4]{5^{8}\times5}=5^{2}\times\sqrt[4]{5}=25\sqrt[4]{5}\approx25\times1.4953 \approx 37.38 \)
Then subtract 2 from both sides:
\[
t=5^{\frac{9}{4}}- 2\approx37.38 - 2=35.38
\]
(We can also solve it more precisely using logarithms. Let's redo step 2 - 4 with more precise steps. Starting from \( \log_{5}(t + 2)=\frac{9}{4} \), we can use the change of base formula. \( \log_{5}(t + 2)=\frac{\ln(t + 2)}{\ln(5)}=\frac{9}{4} \), so \( \ln(t + 2)=\frac{9}{4}\ln(5) \), then \( t + 2 = e^{\frac{9}{4}\ln(5)}=5^{\frac{9}{4}} \), which is the same as before. Alternatively, we can use the original equation and solve step by step. Let's check the calculation of \( 5^{\frac{9}{4}} \):

\( 5^{2}=25 \), \( 5^{\frac{1}{4}}\approx1.4953 \), so \( 25\times1.4953 = 37.3825 \), then \( t=37.3825 - 2=35.3825 \approx 35.38 \))

Answer:

The video will take approximately \( \boldsymbol{35.4} \) days (or more precisely around 35.38 days) to be viewed 50,000 times.