Problem 64 - College Algebra

Question

The exposure index EI for a 35 millimeter camera is a measurement of the amount of light that hits the film. It is determined by the equation \(E I=\log _{2}\left(\frac{f^{2}}{t}\right),\) where \(f\) is the "f-stop" setting on the camera, and \(t\) is the exposure time in seconds. Suppose the f-stop setting is 8 and the desired exposure time is 2 seconds. What will the resulting exposure index be?

Step-by-Step Solution

Answer

The exposure index EI is 5.

Step 1: Identify Known Variables

In the given problem, you are provided with a few variables. The f-stop setting is \( f = 8 \) and the exposure time is \( t = 2 \) seconds.

Step 2: Substitute Known Values into the Equation

The formula for the exposure index is \( EI = \log_{2}\left(\frac{f^{2}}{t}\right) \). Substitute \( f = 8 \) and \( t = 2 \) into the formula:\[ EI = \log_{2}\left(\frac{8^{2}}{2}\right) \]

Step 3: Simplify Inside the Logarithm

First, calculate \( 8^{2} \), which is \( 64 \). Then divide by 2:\[ \frac{64}{2} = 32 \].So, the equation becomes \[ EI = \log_{2}(32) \].

Step 4: Calculate the Logarithm Base 2

Find \( EI = \log_{2}(32) \). Recognize that 32 is \( 2^5 \), so we have \( \log_{2}(2^5) \):Since \( \log_{b}(b^x) = x \), we have \[ \log_{2}(32) = 5 \].

Step 5: Conclusion

Therefore, the exposure index \( EI \) is 5.

Key Concepts

Camera Exposure Settings

Understanding camera exposure settings is crucial for capturing well-lit images. Exposure is the amount of light that reaches your camera sensor, and it directly affects how bright or dark an image will be. Three main elements control this exposure: the f-stop, exposure time, and ISO. In this context, we'll focus mainly on the f-stop and exposure time.

The f-stop, also known as the aperture, controls how much light enters through the lens. A lower f-stop number means a larger aperture, allowing more light to hit the sensor.

• Larger apertures (small f-stop numbers) let in more light, suitable for low-light conditions.
• Smaller apertures (high f-stop numbers) provide a larger depth of field, useful for landscape photography.

Exposure time, or shutter speed, refers to how long the camera's shutter remains open to let light in. A longer exposure time lets more light hit the sensor but can also introduce blur if the subject is moving. Students should remember:

• Short exposure time: Less light enters, good for fast-moving subjects.
• Long exposure time: More light enters, useful in dim lighting but may cause motion blur.

Mastering these settings helps photographers achieve the right balance between light and image clarity.

F-stop and Exposure Time

The f-stop and exposure time are two of the critical settings in camera photography that influence the exposure index (EI). The f-stop number describes the size of the aperture, while the exposure time dictates how long the camera sensor is exposed to light.

In the equation for the exposure index given in the problem, \[EI = \log_{2}\left(\frac{f^{2}}{t}\right),\]f represents the f-stop, and t symbolizes the exposure time. Adjusting these can dramatically change the image's exposure.

• A lower f-stop can result in a brighter image, as more light is allowed into the camera.
• Increasing the exposure time also allows more light to accumulate, brightening the image further.

In the example problem, using an f-stop of 8 and an exposure time of 2 seconds, you determine the EI by substituting these values into the given equation, illustrating how these components work together to affect exposure.

Mathematical Problem-Solving

Mathematical problem-solving is an essential skill that applies to many real-life scenarios, including photography. Solving the given exercise involves several straightforward steps that demonstrate how to apply mathematical logic to a practical problem.

First, identify the known variables. In our example, the f-stop is 8 and the exposure time is 2 seconds. Next, substitute these values into the relevant equation to find the exposure index.

The equation is \[EI = \log_{2}\left(\frac{f^{2}}{t}\right).\]Here's a breakdown of the solution steps:

• Calculate \(f^{2}\) which is \(8^{2} = 64\).
• Divide by the exposure time, \(t = 2\), resulting in \( \frac{64}{2} = 32\).
• Calculate \( \log_{2}(32)\). Recognize that 32 equals \(2^5\), so \( \log_{2}(2^5) = 5\).

Through this method, students can see the relationship between mathematical equations and practical applications like camera settings. This exercise reinforces the importance of breaking down problems into manageable parts and applying logical reasoning to reach a solution.