Problem 37 - Algebra and Trigonometry

Question

Use a calculator to evaluate the expression, correct to four decimal places. $$ \begin{array}{llll}{\text { (a) } \log 2} & {\text { (b) } \log 35.2} & {\text { (c) } \log \left(\frac{2}{3}\right)}\end{array} $$

Step-by-Step Solution

Answer

(a) 0.3010, (b) 1.5465, (c) -0.1761

Step 1: Evaluating Logarithm of 2

Using a calculator, input \( \log 2 \) to find its logarithm to four decimal places. The value is approximately \( 0.3010 \).

Step 2: Evaluating Logarithm of 35.2

For \( \log 35.2 \), enter this into your calculator. The logarithmic value you should get is approximately \( 1.5465 \).

Step 3: Evaluating Logarithm of a Fraction

Input the fraction into the calculator as \( \log \left( \frac{2}{3} \right) \). The result is approximately \( -0.1761 \).

Key Concepts

Evaluating Logarithms

When we talk about evaluating logarithms, we mean finding the numerical value of the logarithm for a given number. Logarithms are the inverse operation of exponentiation. Let's break this down in simple terms:

• The logarithm of a number is the exponent to which the base must be raised to obtain that number.
• The most common bases are 10 (common logarithms) and e (natural logarithms), but any positive number can be a base.

To evaluate logarithms using a calculator, like in the exercise, you'll usually input the number and press the "log" button. This button typically calculates the logarithm for the base 10. For instance, in the steps given for the exercise:

• For \( \log 2 \), you simply enter 2 and press "log" on your calculator, obtaining approximately 0.3010.
• Likewise, for \( \log 35.2 \), the calculator gives around 1.5465. These results are exact up to four decimal places.

Using a calculator simplifies this process and ensures precision.

Logarithmic Functions

A logarithmic function is a function that uses a logarithm. If you've ever worked with equations like \( y = \log_b(x) \), where \( b \) is the base of the logarithm, you've encountered a logarithmic function. They have some unique characteristics:

• They are the inverse of exponential functions. This means for the exponential function \( y = b^x \), its inverse would be \( x = \log_b(y) \).
• On a graph, logarithmic functions typically have a distinctive curve that rises slowly, especially at higher values of x.

Logarithmic functions are essential in many areas including scientific calculations and financial computations. They help solve equations where getting the exponent alone is necessary. Understanding the curve and behavior of logarithmic functions can empower you to apply them in various real-world scenarios.

Fractional Logarithms

Fractional logarithms refer to finding the logarithm of a fraction. This might seem a bit tricky at first, but it's quite manageable when you understand the basic property of logarithms: \[\log \left( \frac{a}{b} \right) = \log a - \log b\]This means the logarithm of a fraction can be expressed as the difference of the logarithms of the numerator and the denominator.

• For example, using the exercise's step-by-step solution: \( \log \left( \frac{2}{3} \right) \), we find this approximately at -0.1761.
• This solution was calculated by inputting \( \frac{2}{3} \) into a calculator and using the "log" function, which then applies the above-mentioned property.

Understanding fractional logarithms is crucial in calculus and other advanced mathematics fields because it simplifies the analysis of functions involving fractions.