Problem 38 - Algebra 2

Question

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. $$ \log _{4}(1.6)^{2} $$

Step-by-Step Solution

Answer

0.6776

Step 1: Apply Power Rule

The power rule for logarithms states that \( \log_b(x^n) = n \log_b(x) \). Here, we have \( (1.6)^2 \) inside the logarithm, so by applying the power rule, it becomes \( 2 \log_4(1.6) \).

Step 2: Use Change of Base Formula

The change of base formula for logarithms is \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \). We'll use common logarithms (base 10), so \( \log_4(1.6) = \frac{\log_{10}(1.6)}{\log_{10}(4)} \).

Step 3: Substitute into Expression

Substituting the result from Step 2 into the expression from Step 1 gives us: \[ 2 \log_4(1.6) = 2 \times \frac{\log_{10}(1.6)}{\log_{10}(4)}. \]

Step 4: Calculate Values

Use a calculator to approximate the values of the logarithms. \( \log_{10}(1.6) \approx 0.2041 \) and \( \log_{10}(4) \approx 0.6021 \).

Step 5: Compute the Final Approximation

Plug these values back into the equation from Step 3: \[ 2 \times \frac{0.2041}{0.6021} \approx 2 \times 0.3388 = 0.6776. \]

Key Concepts

Change of Base Formula

Logarithms, like many mathematical concepts, have rules that make them easier to work with. The change of base formula is a very handy tool that allows us to compute logarithms with bases other than 10 or e by using our calculators. Most calculators are limited to base 10 (known as common logarithms) or base e (natural logarithms).

This formula is written as:

• \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \)

Here, \( b \) and \( a \) are any positive numbers, and \( c \) is the base that you choose, like 10 or e. The change of base formula essentially allows you to convert your logarithm to a more convenient base for calculation.

In our exercise, we used the change of base formula to convert \( \log_4(1.6) \) into terms of common logarithms. This made it easier for us to handle in practical calculations, turning it into \( \frac{\log_{10}(1.6)}{\log_{10}(4)} \).

Power Rule for Logarithms

The power rule for logarithms is a simple yet powerful property that helps simplify the log expression when you are dealing with exponential values. This rule states that \( \log_b(x^n) = n \cdot \log_b(x) \).

Whenever you see an exponentiation within a logarithm, you can move that exponent in front of the log, making the calculation more straightforward. This trick is particularly useful when expressions within the log have powers that are easier to handle outside of it.

In the given problem, we had \( \log_4((1.6)^2) \). By applying the power rule, it becomes \( 2 \cdot \log_4(1.6) \). This transformation is crucial as it allows further manipulation, such as applying the change of base formula to convert the expression into common logarithms that a calculator can then evaluate easily.

Common Logarithms

Common logarithms are logarithms with a base of 10. They are often the go-to logarithm type because they are easily computed using most scientific calculators, which usually have a dedicated \( \log \) button for base 10.

In mathematical notation, you will see \( \log(x) \) without any base specified; this usually implies that it is a common logarithm. Logarithms with a base of 10 are very practical in real-world applications, ranging from scientific calculations to financial models, as many natural phenomena are logarithmically based on powers of 10.

In our activity, converting the original log expression to involve common logarithms was a vital step that allowed for easy calculation. By expressing \( \log_4(1.6) \) in terms of common logarithms using the change of base formula, we calculated the common logarithms to approximate its value efficiently. This is why understanding common logarithms, alongside the associated formulae and rules, is crucial for efficiently solving logarithmic problems like the one provided.