Problem 59 - Algebra for College Students

Question

Given that \(\log _{2} 5=2.3219\) and \(\log _{2} 7=2.8074\), evaluate each expression by using Properties \(10.5-10.7\) \(\log _{2} 80\)

Step-by-Step Solution

Answer

\(\log_2 80 = 6.3219\)

Step 1: Express 80 in terms of powers of 2, 5, and 7

First, recognize that 80 can be expressed as the product of smaller numbers whose logarithms we know. We know that 80 is not a power of 2 alone, so we need to break it down further. Observe that \[ 80 = 2^4 imes 5 \]This gives us a combination of powers that fits our known values.

Step 2: Apply the product property of logarithms

Use the logarithmic identity \[ \log_b (xy) = \log_b x + \log_b y \]Thus,\[ \log_2 80 = \log_2 (2^4 \times 5) = \log_2 2^4 + \log_2 5 \].

Step 3: Simplify using known logarithm values

Use the identity \[ \log_b (b^n) = n \log_b b = n \].Thus,\[ \log_2 2^4 = 4 \cdot \log_2 2 = 4 \cdot 1 = 4 \].Now the expression becomes\[ \log_2 80 = 4 + \log_2 5 \].

Step 4: Substitute the known logarithm value of \(\log_2 5\)

We are given that \(\log_2 5 = 2.3219\). Substitute this value into the equation:\[ \log_2 80 = 4 + 2.3219 = 6.3219 \].

Key Concepts

Product Property of Logarithms

Logarithms have some very useful properties, and one of the key ones is the product property. This property states that the logarithm of a product is equal to the sum of the logarithms of each individual factor. Mathematically, this is expressed as: \[ \log_b (xy) = \log_b x + \log_b y \] This can be an enormous help when evaluating more complex logarithmic expressions.

For example, if you need to find \( \log_2 80 \), rather than trying to find the log of 80 directly, you can break 80 into its factors to make the computation simpler. Since 80 can be factored into \( 2^4 \times 5 \), you can then use the product property to find \( \log_2 80 = \log_2 (2^4 \times 5) = \log_2 2^4 + \log_2 5 \). This approach makes use of known values and simplifies the calculation process.

Logarithmic Identities

Logarithmic identities are like handy shortcuts that predictably describe how logarithms behave. These identities help students navigate through problems by reducing complexity into simpler computations.

One important identity is the power rule, expressed as: \[ \log_b (b^n) = n \log_b b = n \] This identity says that for a base \( b \), if the number is a power of that base, you can simply multiply the exponent \( n \) by the log of the base itself. In the case of evaluating \( \log_2 80 \), this identity allows us to simplify \( \log_2 2^4 \). Since \( \log_2 2 = 1 \), it follows that \( 4 \cdot \log_2 2 = 4 \cdot 1 = 4 \).

These identities streamline logarithmic calculations, enabling solutions that might otherwise take much longer if attempted through basic repeated multiplication or division.

Logarithmic Evaluation

Evaluating logarithms often requires using known values, especially when working with complex numbers. This process involves substituting the known log values into the equations after simplifying them using logarithmic properties and identities.

In our exercise, we need to find \( \log_2 80 \). After breaking down 80 into \( 2^4 \times 5 \) and applying the product and power properties, the expression simplifies to \( 4 + \log_2 5 \). We substitute the given \( \log_2 5 = 2.3219 \) into this equation.

• First, use the power property: \( \log_2 2^4 = 4 \)
• Then, use the known value: \( \log_2 80 = 4 + 2.3219 \)

Through this method, we find \( \log_2 80 = 6.3219 \).

This process shows that with a systematic approach and utilizing known values, solving complex logarithmic evaluations becomes manageable and straightforward.