Problem 71 - College Algebra with Modeling and Visualization

Question

Use the change of base formula to approximate the logarithm to the nearest thousandth. $$ \log _{2} 5+\log _{2} 7 $$

Step-by-Step Solution

Answer

\( \log _{2} 5+\log _{2} 7 \approx 5.130 \)

Step 1: Use Properties of Logarithms

Recall that the sum of two logarithms with the same base can be written as a single logarithm by the property \( \log_b(m) + \log_b(n) = \log_b(mn) \). Hence, \( \log_2(5) + \log_2(7) = \log_2(5 \times 7) = \log_2(35) \).

Step 2: Apply the Change of Base Formula

The change of base formula allows us to evaluate logarithms with any base by converting them to common logarithms. It states \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \). Choose base 10 (common logarithm), so \( \log_2(35) = \frac{\log_{10}(35)}{\log_{10}(2)} \).

Step 3: Calculate Logarithms in Base 10

Use a calculator to find \( \log_{10}(35) \approx 1.544 \) and \( \log_{10}(2) \approx 0.301 \, 0.301 \).

Step 4: Compute the Division

Substitute the values found in Step 3 into the expression to obtain \( \log_2(35) \approx \frac{1.544}{0.301} \). Upon calculation, this yields approximately 5.130.

Step 5: Round to the Nearest Thousandth

Since the result from Step 4 is 5.130, it is already at thousandth level precision, so no further rounding is needed.

Key Concepts

Change of Base Formula

The change of base formula is a powerful tool when dealing with logarithms, especially when your calculator does not support certain logarithmic bases. This formula allows you to convert a logarithm of any base into a base that is more easily computed, such as base 10 or base e (the natural logarithm). The formula is expressed as \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \). Here, \( b \) is the original base of the logarithm, \( a \) is the argument, and \( c \) is the new base you're converting to.

Using these bases helps simplify the computation process significantly. For example, when you need to solve \( \log_2(35) \), you convert it using the formula into \( \frac{\log_{10}(35)}{\log_{10}(2)} \). In this example, "10" is used for the common logarithm, making it easier to evaluate using multiplication or division features on calculators.

Once converted, the calculation becomes straightforward, just needing you to find the common logarithms of 35 and 2, then divide the former by the latter.

Properties of Logarithms

Understanding properties of logarithms is crucial when simplifying expressions before calculation. One key property is the product rule, \( \log_b(m) + \log_b(n) = \log_b(mn) \). This property allows you to combine the sum of two logarithms into a single logarithm. In our example, \( \log_2(5) + \log_2(7) \) was simplified to \( \log_2(35) \).

Other properties to keep in mind include the power rule: \( \log_b(m^n) = n\log_b(m) \), and the quotient rule: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \).

These properties make it easier to manipulate logarithmic expressions, helping to simplify complex expressions and find solutions more efficiently.

Common Logarithms

Common logarithms are logarithms with base 10. They are widely used in a variety of applications, from scientific calculations to computing ease, due to the straightforwardness in calculations; most calculators have a "log" button which calculates \( \log_{10}(x) \).

In the example, once \( \log_2(35) \) was converted using the change of base formula, we used common logarithms to find \( \log_{10}(35) \approx 1.544 \) and \( \log_{10}(2) \approx 0.301 \). These calculations allow us to use easily available tools to compute values that would otherwise be complicated.

Sometimes, common logarithms are simply referred to as "log" without specifying the base, since base 10 is implied. They simplify significant digits and making mathematical work more efficient overall.