Problem 20 - Algebra 2 and Trigonometry

Question

In \(15-23,\) evaluate each logarithm to the nearest hundredth. $$ 200 \log \frac{5}{2} $$

Step-by-Step Solution

Answer

The value is approximately 79.59.

Step 1: Understand the Expression

You need to evaluate the expression \(200 \log \frac{5}{2}\). This involves calculating the logarithm of a fraction and then multiplying the result by 200.

Step 2: Calculate the Fraction

Find the fraction \(\frac{5}{2}\), which remains the same for logarithmic calculations. The decimal equivalent of \(\frac{5}{2}\) is 2.5.

Step 3: Use the Logarithm Function

Use your calculator to find \(\log 2.5\). On most calculators, you go to the log function and enter 2.5. This gives the base 10 logarithm of 2.5, which is approximately 0.39794.

Step 4: Multiply by 200

Multiply the logarithm found in Step 3 by 200: \[200 \times 0.39794 = 79.588\]Round this to the nearest hundredth to get 79.59.

Key Concepts

Logarithm of a Fraction

A logarithm is a mathematical function that helps us find the power to which a number, called a base, must be raised to produce a given number. When you deal with fractions in logarithms, like in our problem which involves \( \log \frac{5}{2} \), you compute the logarithm of the resulting decimal of the fraction.
Fractions are often converted to a decimal form by dividing the numerator by the denominator, making the calculation simpler. In our problem, \( \frac{5}{2} \) converts to 2.5, and we then find the logarithm of 2.5.
Understanding the fraction helps us prepare for the subsequent calculations and ensures we're accurately evaluating the expression.

Base 10 Logarithm

The term "log" is commonly used to refer to the base 10 logarithm, also known as the "common logarithm." It is one of the most frequently used logarithms in mathematics. This concept refers to applying the logarithm function with a base of 10.
For example, when you calculate \( \log 2.5 \), it implies that you are using a base 10. Essentially, you're trying to determine what power 10 must be raised to, to result in 2.5.
On most scientific calculators, the "log" button signifies the base 10 logarithm. Just input the number 2.5 to find the result, which is approximately 0.39794 in this specific exercise.

Rounding Numbers

Rounding numbers is important for simplifying numerical results, especially when they come from more complex calculations. In our exercise, after calculating the expression \( 200 \times 0.39794 \), we reach the result \( 79.588 \).
To round this number to the nearest hundredth, we must focus on the digits following the decimal point. Keep the first two decimal digits and look at the third. If it is 5 or greater, round up; if less, keep the first two digits as they are.
Therefore, since the third digit (8 in 79.588) is greater than 5, we round up to 79.59.

Logarithmic Calculations

Logarithmic calculations help solve a variety of mathematical problems, providing insights into geometric growth, scientific measurements, and financial calculations. The exercise involves multiplying the resulting logarithm by another number, which here is 200.
First, find the logarithm of a given number, then use simple multiplication to adjust the scale of impact the logarithm has, as seen with \( 200 \log \frac{5}{2} \).
This procedure enhances the logarithm’s effect, interpreting the logarithm not just as a descriptor of a single number's base power, but as a scaled representation, helping us understand larger numerical transformations or predict outcomes.