Problem 40 - Algebra for College Students

Question

Approximate each logarithm to three decimal places. $$ \log _{5} 0.047 $$

Step-by-Step Solution

Answer

\( \log_{5} 0.047 \approx -1.898 \).

Step 1: Apply Change of Base Formula

To compute \( \log_{5} 0.047 \), use the change of base formula: \( \log_{a} b = \frac{\log_{10} b}{\log_{10} a} \). Therefore, \( \log_{5} 0.047 = \frac{\log_{10} 0.047}{\log_{10} 5} \).

Step 2: Compute \( \log_{10} 0.047 \)

Calculate \( \log_{10} 0.047 \) using a calculator. It is approximately \( -1.327 \) (rounded to three decimal places).

Step 3: Compute \( \log_{10} 5 \)

Calculate \( \log_{10} 5 \) using a calculator. It is approximately \( 0.699 \) (rounded to three decimal places).

Step 4: Divide the Results

Apply the division from Step 1: \( \frac{-1.327}{0.699} = -1.898 \).

Step 5: Approximated Result

Upon calculating and rounding to three decimal places, \( \log_{5} 0.047 \approx -1.898 \).

Key Concepts

Change of Base Formula

The change of base formula is a powerful tool when working with logarithms. Often, we deal with uncommon or difficult bases, like base 5 in the example problem. This formula allows us to convert a logarithm from one base to another, often making it easier to solve.

The formula is expressed as:

• \( \log_{a} b = \frac{\log_{10} b}{\log_{10} a} \)

This formula means that a logarithm with an awkward base can be rewritten as a ratio of two logarithms in base 10, which is more standard and easier to calculate. Base 10 logarithms, known as common logarithms, are available on most calculators.

In the formula, \(a\) and \(b\) are the original numbers in the logarithm calculation, and you are transforming it to use the base 10 logarithms. This makes the change of base formula extremely useful for simplifying calculations in mathematics.

Logarithmic Approximation

Logarithmic approximation is crucial when solving logarithms without exact solutions or when using tools that provide decimal places, like calculators. Often, we're required to find the logarithm of an exact value but only need it to a specific number of decimal places.

Take the provided problem where \( \log_{10} 0.047 \) is calculated and approximated to \(-1.327\). This is achieved using a calculator, which gives a precise value that can be approximated further. The need to approximate is common in mathematics as not all logarithmic values can be expressed neatly or with a finite number of digits.

In practice, this approximation step helps to produce results that are good enough for practical purposes, such as scientific calculations or real-world applications where being precise is less critical than being close to the true value.

Common Logarithms

Common logarithms, or base 10 logarithms, are a standard form in mathematical computations. They are denoted as \( \log_{10} \) or even just \( \log \) when the base is not specified. These logarithms are fundamental because of their simplicity and ease of use with calculators, which almost always support base 10 logarithms directly.

In simpler terms, a common logarithm answers the question: "To what power must 10 be raised to produce a given number?" For instance, \( \log_{10} 100 = 2 \) because \(10^2 = 100\).

The advantage of common logarithms comes from their universality in scientific notation, where powers of 10 are frequent, and in financial calculations, simplifying complex exponential growth or decay problems. They are also integral to the change of base formula, as seen in converting \( \log_{5} 0.047 \) into a common logarithm calculation to ease the solution process.