Problem 64 - College Algebra

Question

The exposure index \(E I\) for a 35 millimeter camera is a measurement of the amount of light that hits the fi \(\mathrm{m}\). 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

EI = 5

Step 1: Substitute Values

Substitute the given values into the equation for the exposure index. We have the f-stop setting \( f = 8 \) and the exposure time \( t = 2 \) seconds. The formula is: \[ EI = \log_{2}\left(\frac{f^{2}}{t}\right) \] Substitute the known values into the formula: \[ EI = \log_{2}\left(\frac{8^{2}}{2}\right) \]

Step 2: Calculate f-stop Squared

Compute \( f^2 \), where \( f = 8 \). \[ f^2 = 8^2 = 64 \] So, the expression becomes: \[ EI = \log_{2}\left(\frac{64}{2}\right) \]

Step 3: Simplify the Fraction

Now, simplify \( \frac{64}{2} \). \[ \frac{64}{2} = 32 \] The expression simplifies to: \[ EI = \log_{2}(32) \]

Step 4: Evaluate the Logarithm

Find the value of \( \log_{2}(32) \). Since \( 2^5 = 32 \), we have: \[ \log_{2}(32) = 5 \]

Step 5: Conclusion

The exposure index \( EI \) with the given f-stop setting and exposure time is \( 5 \).

Key Concepts

Logarithms

Logarithms are the opposite of exponentiation. If you are familiar with powers where you raise a number to a certain exponent, logarithms can help you find that exponent.
In the context of the equation given in our exercise, \( EI=\log_{2}\left(\frac{f^{2}}{t}\right) \), the logarithm with base 2 helps us measure how many times 2 must be multiplied by itself to get the result within the parentheses.
Understanding this is crucial because many algebraic expressions involving logarithms involve reversing the process of exponentiation to find unknown values.
When calculating logarithms manually, always identify the base and ask yourself: "What power must the base be raised to in order to get this number?"

Exposure index calculation

The exposure index is an important concept in photography as it helps determine the amount of light captured on the film. It guides photographers to achieve properly exposed photographs by adjusting the 'f-stop' and exposure time.
The equation provided, \( EI = \log_{2}\left(\frac{f^{2}}{t}\right) \), is a mathematical representation that combines two critical factors that affect exposure: the f-stop, which controls the aperture size or lens opening, and the exposure time, which controls how long the lens is open to light.
In this formula:

• \( f \) stands for the f-stop value, which affects the depth of field and exposure.
• \( t \) is the time in seconds during which light enters the camera.

By substituting these variables into the formula, photographers and students alike can calculate how different settings interact to produce a specific exposure index.

Mathematical substitution

Mathematical substitution is a method where we replace variables in an equation with known values. This technique is pivotal in solving algebra problems and in our problem, it signifies plugging in real-world values to calculate the desired outcome.
Initially, we identified that \( f = 8 \) and \( t = 2 \) in our problem. By substituting these into the equation \( EI = \log_{2}\left(\frac{f^{2}}{t}\right) \), it becomes: \( EI = \log_{2}\left(\frac{8^{2}}{2}\right) \).
This substitution bridges the gap between an abstract mathematical formula and its real-world application by ensuring that all variables are assigned values, permitting further calculation. It simplifies the original problem into manageable parts, making the equation ready for the next steps like simplification and evaluation.

Problem solving steps

Problem-solving steps are the systematic approach we take to solve any mathematical question. In our exercise, they involve breaking down the problem into smaller, more manageable parts. Let's see how it works:

• First, **Substitution**: Insert given values into the formula. We start by placing \( f = 8 \) and \( t = 2 \).
• Next, **Calculation**: Evaluate mathematical components like squaring the f-stop value, here \( 8^2 = 64 \).
• **Simplification**: Reduce expressions, notably \( \frac{64}{2} = 32 \).
• Then, **Evaluation**: Use known values or properties (like \( \log_{2}(32) = 5\) since \( 2^5 = 32 \)) to find log results.
• Finally, **Conclusion**: Confirm the exposure index, which leads to our final answer.

This stepwise breakdown clarifies each stage of your solution path, ensuring you don't miss any details and can follow through confidently.