We are given two functions, $f(x) = log_{10}x$ and $f(x) = -\frac{1}{2}log_{10}x - 5$. We need to determine the transformations that turn the first function into the second function.

1. Problem Description

We are given two functions, $f(x) = log_{10}x$ and $f(x) = -\frac{1}{2}log_{10}x - 5$. We need to determine the transformations that turn the first function into the second function.

2. Solution Steps

We start with the function $f(x) = log_{10}x$.

First, consider the transformation $f(x)$ to $-\frac{1}{2}f(x)$.

Multiplying the function by $-1$ reflects the function across the x-axis.

Multiplying the function by $\frac{1}{2}$ vertically compresses the function by a factor of $\frac{1}{2}$.

Therefore, $-\frac{1}{2}f(x) = -\frac{1}{2}log_{10}x$ means a reflection across the x-axis and a vertical compression by a factor of $\frac{1}{2}$.

Next, consider the transformation $-\frac{1}{2}log_{10}x$ to $-\frac{1}{2}log_{10}x - 5$.

Subtracting 5 from the function results in a vertical translation 5 units down.

Therefore, the transformations are:

1. Reflection about the x-axis.

2. Vertical compression by a factor of $\frac{1}{2}$.

3. Vertical translation 5 units down.

3. Final Answer

Reflection about the x-axis, vertical compression by a factor of $\frac{1}{2}$, vertical translation 5 units down.