Problem 52 - Algebra A Combined Function

Question

Approximate each logarithm to four decimal places. $$ \log _{3} 2 $$

Step-by-Step Solution

Answer

The approximate value of \(\log_3 2\) is 0.6309.

Step 1: Understand the Problem

We need to approximate the value of \( \log_3 2 \) to four decimal places. This represents the power to which the base 3 must be raised to obtain the value 2.

Step 2: Use the Change of Base Formula

We can use the change of base formula to calculate logarithms in bases other than 10 or \( e \):\[\log_b a = \frac{\log_c a}{\log_c b}\]We will use base 10, which is common on calculators, so:\[\log_3 2 = \frac{\log_{10} 2}{\log_{10} 3}\]

Step 3: Calculate \(\log_{10} 2\)

Using a calculator, find \(\log_{10} 2\), which is approximately 0.3010.

Step 4: Calculate \(\log_{10} 3\)

Using a calculator, find \(\log_{10} 3\), which is approximately 0.4771.

Step 5: Divide to Find \(\log_3 2\)

Now, divide the values obtained:\[\log_3 2 = \frac{0.3010}{0.4771} \approx 0.6309\]

Step 6: Round to Four Decimal Places

The result is already rounded to four decimal places, so the approximate value of \(\log_3 2\) is 0.6309.

Key Concepts

Change of Base Formula

The change of base formula is a key concept in understanding logarithms with bases other than 10 or the natural base \(e\). This formula allows us to compute these logarithms using a more familiar base, typically base 10 (common logarithms) or \(e\) (natural logarithms). The formula states that for any positive numbers \(a\), \(b\), and \(c\), where \(b eq 1\) and \(c eq 1\), the logarithm of \(a\) with base \(b\) can be expressed with base \(c\) as:

• \(\log_b a = \frac{\log_c a}{\log_c b}\)

This formula is particularly useful because most calculators are equipped to handle logarithms with base 10 or \(e\), but not others. For example, if you wish to calculate \(\log_3 2\), you can use the change of base formula to convert it into:

• \(\log_3 2 = \frac{\log_{10} 2}{\log_{10} 3}\)

By converting these terms into base 10 logarithms, you can then easily input them into your calculator and solve.

Approximation Techniques

Approximation techniques are essential when working with irrational numbers like logarithms, especially when precise values are hard to compute directly. When we talk about approximating, we are trying to find a number that is close enough to the desired value for practical use without needing an exact result.

• In our context, approximation involves computing numerical estimates of logarithmic values by using formulas and calculators.
• For example, when calculating \(\log_{10} 2\) and \(\log_{10} 3\), we found these values as 0.3010 and 0.4771, respectively, using a calculator which already provides approximations for practical purposes.

The goal is often to achieve a balance between accuracy and efficiency, as perfect precision is impossible with irrational numbers. Hence, rounding is applied to fit your results to the required decimal places.

Calculator Usage

Using a calculator is crucial when working with logarithms, especially when dealing with approximation and the change of base formula. A standard scientific calculator simplifies the computation of logarithms by offering built-in functions such as "log" for base 10 and "ln" for base \(e\).

• To calculate \(\log_{10} a\), simply input the number \(a\) and then press the "log" button.
• Similarly, for natural logarithms, \(\ln a\), press the "ln" button on your calculator with your number.

Calculators ensure quick calculations, minimizing manual errors. Ensure your calculator is set to the correct mode to avoid erroneous outputs.
For logarithms with bases other than 10 or \(e\), like \(\log_3 2\), use the change of base formula as mentioned earlier, and input the results step by step into your calculator for accurate approximation.

Math Rounding Rules

Math rounding is the process of adjusting the number to be more convenient or fit to the required precision. Properly rounding numbers makes them easier to work with while providing a close approximation to the true value, particularly in mathematical computations like logarithms.

• If the first digit after the required decimal places is 5 or more, you round up.
• If it is less than 5, you round down.

In our example, \(\log_3 2 = 0.630872...\), rounds to 0.6309 when rounded to four decimal places. This step ensures that our calculation is precise enough for practical purposes without overwhelming complexity.
Accurate rounding is key in many fields requiring precise measurements and results, ensuring consistency of simple yet effective approximation.