Problem 55 - Algebra 2

Question

Assume that \(\log 4 \approx 0.6021, \log 5 \approx 0.6990,\) and \(\log 6 \approx 0.7782 .\) Use the properties of logarithms to evaluate each expression. Do not use your calculator. \(\log \sqrt{5}\)

Step-by-Step Solution

Answer

The value of \(\log \sqrt{5}\) approximately equals to 0.3495

Step 1: Formulate the logarithm expression

Firstly, express the square root in terms of an exponent: \(\log \sqrt{5}\) = \(\log 5^{1/2}\).

Step 2: Apply logarithm property

By using logarithm property, \(\log_a^n x = n * \log_a x\), we get: \(1/2 * \log 5\).

Step 3: Calculate the expression

Now, using the value of \(\log 5 \approx 0.6990\), we find the result by multiplying it by 1/2: \(0.5 * 0.6990 = 0.3495\).

Key Concepts

Properties of Logarithms

Logarithms are powerful mathematical tools that simplify the computation of products, quotients, and powers, among other things. Understanding their properties is essential. One fundamental property is the **product property**, which states that the logarithm of a product is the sum of the logarithms:
\[\log_b (mn) = \log_b m + \log_b n\]This property helps when dealing with unwieldy multiplication inside a log.
Additionally, the **quotient property** tells us that the log of a quotient is equal to the difference of the logs:
\[\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n\]This can be particularly useful for breaking apart divisions inside a log expression.
Lastly, the **power property** indicates that the logarithm of a power is the exponent times the logarithm:
\[\log_b (m^n) = n \cdot \log_b m\]This is crucial for simplifying expressions with exponents, just like in our example where \( \log \sqrt{5} = \log 5^{1/2} = \frac{1}{2} \log 5\).
These properties are foundational, enabling us to manipulate and simplify complex logarithmic expressions without a calculator.

Calculating Logarithmic Expressions

When calculating logarithmic expressions, it's beneficial to express them in a form that allows us to apply known values easily.
In the original exercise, we started with \(\log \sqrt{5}\). By expressing this as \(\log 5^{1/2}\), we transformed the expression into a form where we could apply the power property of logarithms.
This changes it to \(\frac{1}{2} \cdot \log 5\).
Knowing that \(\log 5 \approx 0.6990\), we proceed by multiplying:

• First, rewrite the logarithm using the power property: \(\log 5^{1/2} = \frac{1}{2} \cdot \log 5\)
• Substitute the known value: \(\frac{1}{2} \cdot 0.6990\)

Performing the calculation, we get \(0.3495\), providing a clear and concise evaluation of the log expression without a calculator.
Breaking down expressions in this way is crucial for accuracy and understanding.

Logarithmic Approximation

Logarithmic approximation involves estimating logarithmic values using known benchmarks. Such approximations are especially handy when no calculator is available.
In our exercise, precise values like \(\log 4 \approx 0.6021\), \(\log 5 \approx 0.6990\), and \(\log 6 \approx 0.7782\) serve as these key reference points.
By leveraging these approximate values, we can gauge the value of less straightforward expressions.
When we calculated \(\log \sqrt{5}\), we approximated it as \(0.5 \times 0.6990\), yielding 0.3495 as an approximate value.

• First, simplify the logarithmic expression using known properties.
• Next, apply approximate values to provide an estimation.
• Finally, perform simple calculations to reach the result.

Through this technique, a deeper understanding of approximation with logarithms emerges, showcasing how mathematical principles and estimations converge for problem-solving.