Problem 45 - 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. \(\log 30\)

Step-by-Step Solution

Answer

\(\log 30 \approx 1.4772\)

Step 1: Factorize the number

Break down the number 30 into factors whose logarithm values are given. The numbers 4,5 and 6 are given so we consider these numbers and see how they can be multiplied or divided to get 30. In particular, \(30 = 5*6\).

Step 2: Apply the Multiplication Property of Logarithms

According to this property, the logarithm of a product equals the sum of the logarithms of the factors. In symbols: \(\log mn = \log m + \log n\). So, we can write: \(\log 30 = \log (5 * 6) = \log 5 + \log 6\).

Step 3: Substitute given logarithm values into the Expression

Now, substitute the given values \(\log 5 \approx 0.6990\) and \(\log 6 \approx 0.7782\) into the expression \(\log 30 = \log 5 + \log 6\), which gives \(\log 30 \approx 0.6990 + 0.7782\).

Step 4: Compute Sum

Finally, add up the decimal values to get the approximate result of \(\log 30\). The final answer is found to be approximately \(0.6990 + 0.7782 = 1.4772\).

Key Concepts

Logarithmic Expressions

Logarithmic expressions allow us to simplify complex calculations of exponential quantities. A logarithm essentially tells us the power to which a number, called the base, must be raised to obtain another number. For instance, \(\log_b a\) tells us what power \(b\) needs to be raised to in order to get the number \(a\). This might sound tricky at first, but it's a powerful tool.

When you see an expression like \(\log 30\), it represents the logarithm base 10 of 30. In many exercises, you're asked to compute this value using known values of logarithms for other numbers, like 4, 5, and 6. This is where the properties of logarithms become extremely helpful. They allow us to break down and manipulate these expressions without needing a calculator.

Understanding how to express a number using another set of numbers (like breaking down 30 into 5 and 6) is a key step in solving logarithmic expressions.

Multiplication Property of Logarithms

Among the most useful properties of logarithms is the multiplication property. This states that the logarithm of a product is equal to the sum of the logarithms of each factor. Mathematically, it is represented as: \(\log (mn) = \log m + \log n\).

This property allows us to transform a complex logarithmic expression into simpler ones by breaking down the number into a product of its factors. For the exercise \(\log 30\), knowing that 30 can be factored into 5 and 6, we can rewrite it as \(\log 5 + \log 6\).

This technique is invaluable when dealing with larger numbers, especially when specific logarithmic values are known. It simplifies calculations and provides a clear path to finding an approximate answer using basic arithmetic.

Logarithm Approximation

Logarithm approximation is a technique used to find the logarithm of a number without a calculator by using known values and properties. In many practical situations, exact values of logarithms are not available, and approximation provides a sensible estimation.

In the given exercise, instead of calculating \(\log 30\) directly, we use known approximations: \(\log 5 \approx 0.6990\) and \(\log 6 \approx 0.7782\). By applying the multiplication property and substituting these known values, we find that \(\log 30 \approx 0.6990 + 0.7782 = 1.4772\).

This approach is particularly useful for exercises or situations where a rough estimate suffices. It demonstrates how methodically applying properties and known values can simplify complex problems into basic arithmetic operations, giving students a powerful tool in mathematical problem-solving.