Problem 186 - Precalculus

Question

The exposure index \(E I\) for a 335 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: Substitute Given Values

We start by substituting the given values into the formula for exposure index: \(EI = \log_2\left(\frac{f^2}{t}\right)\). Here, \(f = 8\) and \(t = 2\). Thus, the equation becomes \(EI = \log_2\left(\frac{8^2}{2}\right)\).

Step 2: Calculate the Fraction

Compute \(\frac{8^2}{2}\). Here, \(8^2 = 64\), so the fraction becomes \(\frac{64}{2}\), which equals 32.

Step 3: Apply Logarithm

Find \(\log_2(32)\). Since \(32 = 2^5\), \(\log_2(32) = 5\).

Step 4: Conclusion

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

Key Concepts

f-stop

The term "f-stop" refers to the aperture setting of a camera lens. It determines the size of the lens opening, which directly affects how much light is allowed to hit the film or sensor. The f-stop settings are usually presented in a sequence, such as 2, 2.8, 4, 5.6, 8, 11, 16, etc.

Every step in this sequence is referred to as "one stop" and represents either a doubling or halving of the light entering the lens. A lower f-stop number means a larger aperture, allowing more light to pass through. Conversely, a higher f-stop number means a smaller aperture, letting in less light.

This adjustment of the aperture affects not only the exposure but also the depth of field in a photograph:

• A lower f-stop results in a shallow depth of field, where the background might be blurred.
• A higher f-stop gives a deeper depth of field, keeping more of the scene in focus.

In the given problem, an f-stop of 8 is used, which is relatively small, thus permitting less light compared to a lower f-stop setting.

exposure time

Exposure time, often referred to as "shutter speed," is the length of time for which the camera's shutter is open to allow light onto the camera sensor or film. It's measured in seconds or fractions of a second, such as 1/1000, 1/500, 1/250, all the way to several seconds.

Longer exposure times let more light onto the film, useful in low-light conditions or when you want to capture motion blur. Shorter exposure times are ideal for freezing motion, preventing bright conditions from overexposing the photo.

In photography, finding the right balance between exposure time and other settings like f-stop and ISO is essential for achieving the desired photograph quality:

• A fast shutter speed (short exposure time) might require a larger aperture to maintain adequate exposure.
• A slow shutter speed (long exposure time) might benefit from a smaller aperture to avoid overexposure.

For the situation outlined in the exercise, the exposure time is set to 2 seconds. It's relatively long, meaning the f-stop must be adjusted accordingly to get the correct exposure.

logarithm base 2

Logarithms are a way to express relationships in terms of powers or exponents. In this context, we use a logarithm with a base of 2, which is often denoted as \(\log_2\). This type is popular in computing and information theory.

The expression \(\log_2(x)\) answers the question, “To what power must 2 be raised, to obtain \(x\)?” For example, \(\log_2(8) = 3\) because \(2^3 = 8\).

In photography exposure, using a base-2 logarithm helps make calculations easy when dealing with doubling or halving light amounts, which aligns with how cameras adjust their settings:

• Each increment in \(\log_2\) corresponds to a doubling of the light exposure.
• This relationship is why f-stop and exposure time settings are structured the way they are, allowing precise control over the light.

In the original exercise, the exposure index \(EI\) is derived from \(\log_2\) of a ratio, effectively capturing how both the aperture size and exposure duration contribute to the light entering the camera.