Problem 90 - Introductory and Intermediate Algebra for College Students 4th

Question

a. Evaluate the expression in part (a) without using a calculator. b. Use your result from part (a) to write the expression in part (b) as a single logarithm whose coefficient is 1 a. \(\log _{2} 16\) b. \(\log _{2} x+5 \log _{2} y-4\)

Step-by-Step Solution

Answer

For part (a), the answer is 4. For part (b), the expression as a single logarithm is \( \log _{2} 16xy^5 \).

Step 1: Evaluate the logarithm in Part (a)

The expression given in part (a) is \( \log _{2} 16 \). Recall that a logarithm is the inverse operation to exponentiation. So, if \( a^b = c \), then \( \log _{a} c = b \). So, thinking about how to express 16 as 2 to some power, we quickly realize \( 2^4 = 16 \). Therefore, \( \log _{2} 16 = 4 \).

Step 2: Prepare for Part (b)

Our result from part (a) will help us write an expression as a single logarithm. We now move on to part (b) which uses the expression \( \log _{2} x+5 \log _{2} y-4 \). We must remember that we are looking to rewrite this expression as a single logarithm with a coefficient of 1.

Step 3: Write the expression as a single logarithm

Using the properties of logarithms, we can break this down as such. First, we recognize that multiplication inside the log can turn into addition outside, so \( 5\log _{2}y = \log _{2} y^5 \). Similarly, exponentiation inside the log can turn into subtraction outside, so \(-4 = \log _{2} (2^{-4}) \). Therefore, we can write our expression as \( \log _{2} x + \log _{2} y^5 - \log _{2} (2^{-4}) \). Since when subtracting logs we divide the entities inside the logs and when adding we multiply them, our final expression is \( \log _{2} [x * y^5 / (2^{-4})] = \log _{2} 16xy^5 \).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential for simplifying logarithmic expressions and solving logarithmic equations. These properties are based on the fundamental definition of logarithms as inverses of exponentiation. Let's explore these properties using simple language and examples.