Problem 68 - Algebra 2

Question

Express each logarithm in terms of common logarithms. Then approximate its value to four decimal places. \(\log _{6} 0.047\)

Step-by-Step Solution

Answer

The approximate value of \( \log_6 (0.047) \) is \(-1.7069\).

Step 1: Understand the Change of Base Formula

To convert a logarithm with a base other than 10 or e to a common logarithm (base 10), you use the change of base formula. The formula is: \[ \log_b (a) = \frac{\log_c (a)}{\log_c (b)} \]For this exercise, we will be using common logarithms (logarithms with base 10). Thus our formula becomes:\[ \log_6 (0.047) = \frac{\log_{10} (0.047)}{\log_{10} (6)} \]

Step 2: Apply the Change of Base Formula

Substitute the values into the change of base formula:\[ \log_6 (0.047) = \frac{\log_{10} (0.047)}{\log_{10} (6)} \]This expression represents the logarithm \( \log_6 (0.047) \) in terms of common logarithms.

Step 3: Calculate the Logarithms Using a Calculator

Now, calculate the values using a calculator:1. Find \( \log_{10} (0.047) \). We get approximately \(-1.3279\).2. Find \( \log_{10} (6) \). We get approximately \(0.7782\).Now, substitute these values into the expression.

Step 4: Solve the Expression

Substitute the logarithm values into the change of base formula:\[ \log_6 (0.047) = \frac{-1.3279}{0.7782} \]Calculate this expression to find the approximate value.

Step 5: Final Calculation

Perform the division:\[ \frac{-1.3279}{0.7782} \approx -1.7069 \]So, the approximate value of \( \log_6 (0.047) \) in terms of common logarithms is \(-1.7069\).

Key Concepts

Logarithms

Logarithms are a crucial concept in mathematics that relate exponential and multiplicative operations. They are essentially the inverse operation of exponentiation. When we hear the term "logarithm," it refers to the power to which a certain base is raised to obtain a number. For example, if we have a logarithm \( \log_b(a) \), it means the power to which the base \( b \) must be raised to get the number \( a \). Simply put, \( b^x = a \) translates into \( x = \log_b(a) \).

• Logarithms help to solve equations involving exponents.
• They simplify complex multiplication and division operations into addition and subtraction.
• Common applications include scientific calculations, sound intensity, and data analysis.

Understanding how to manipulate and use logarithms is essential in various branches of science and engineering, as they provide a means to handle very large numbers and growth rates.

Common Logarithms

Common logarithms are logarithms with base 10, and they are the most frequently used in everyday applications. Mathematically, they are denoted as \( \log_{10}(a) \), or simply \( \log(a) \). This is because humans generally think in base 10, given our ten fingers make this natural for us.

• Common logarithms simplify the calculation process, especially when using older logarithm tables or basic calculators.
• They are popular in chemistry and biology for calculations involving pH values and exponential growth.
• These logarithms also provide a simple way to express exponential data, such as population growth and radioactive decay.

To understand their applications better, consider the following: if you are calculating the common logarithm of a number like 1000, you are finding the power to which 10 must be raised to give you 1000. The answer, of course, is 3 since \( 10^3 = 1000 \). They are especially useful when applying the change of base formula.

Base Conversion

Base conversion is a mathematical technique used to change the base of a logarithm from one number to another. This is particularly helpful when you're dealing with numbers whose base isn't convenient for calculation. One of the keys to base conversion is the Change of Base Formula:\[\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\]When dealing with bases like 6 in \( \log_6(0.047) \), this formula helps convert it into a base 10 logarithm, which is easier to compute:

• You take the logarithm of your number and divide it by the logarithm of the old base, all in terms of the new base.
• This is extremely useful in scenarios where calculators or tools only support common or natural logarithms.
• By converting the base, it becomes feasible to solve complex logarithmic expressions using simple tools.

In our original exercise, using the change of base formula allows us to express \( \log_6(0.047) \) in terms of common logarithms, making it straightforward to estimate using a standard calculator.