Problem 68 - Algebra 2

Question

Write each logarithm as the quotient of two common logarithms. Do not simplify the quotient. $$ \log _{7} 2 $$

Step-by-Step Solution

Answer

\(\log_7 2 = \frac{\log 2}{\log 7}\)

Step 1: Understand the change of base formula

The change of base formula states that for given base \(b\), and numbers \(n\) and \(a\): \[ \log_b a = \frac{\log n a}{\log n b} \] This formula allows us to change the base of a logarithm while keeping its value the same.

Step 2: Apply the formula to the given problem

We are given \(\log_7 2\) and we want to express it as a quotient of two common logarithms. Common logarithms have base 10, hence, according to the change of base formula, the given log can be written as: \[ \log_7 2 = \frac{\log 2}{\log 7} \]

Key Concepts

Understanding Logarithms

Logarithms are mathematical expressions that help us understand the concept of powers and exponents better. They are the opposite operation of exponentiation, just like subtraction is the opposite of addition. Simply put, if you know the exponential expression \[ b^x = a, \] then the logarithmic expression associated with it would be \[ \log_b a = x. \] This means that the logarithm \( \log_b a \) answers the question: 'To what power must the base \( b \) be raised, to produce \( a \)?'
Logarithms are widely used in many fields like science, engineering, and even music. They make it easier to handle very large numbers or very small fractions by converting them into simpler forms.

Exploring Common Logarithms

Common logarithms are the logarithms that use 10 as their base. In mathematical notation, common logarithms are written as \( \log_{10} \). However, it's common practice to omit the base 10 and just write \( \log \) for simplicity. This shorthand helps in calculations and keeps equations looking tidy.
Common logarithms come into play when dealing with real-world problems, especially those involving magnitudes such as scientific data measurements and sound intensity. The common logarithm simplifies complex calculations with these types of data.
For example, the Richter scale for earthquakes is logarithmic, which means each whole number increase on the scale represents a tenfold increase in measured amplitude and approximately 31.6 times more energy release.

Decoding Base Conversion

Base conversion is an essential tool in mathematics, especially when dealing with logarithms of bases other than 10. The change of base formula allows us to convert logarithms to a desired base, and is particularly useful for converting to common logarithms or natural logarithms, where the base is \( e \).
The formula for changing the base of a logarithm is: \[ \log_b a = \frac{\log_n a}{\log_n b}, \] where \( n \) is the new base you wish to convert to. To express logarithms as common logarithms, \( n \) would be 10.

• This formula provides flexibility in mathematical computations.
• It maintains the equality of an equation by balancing the change of base in both the numerator and the denominator.
• Using common logarithms makes it easier to solve calculations that involve non-standard bases.

Base conversion ensures you can continue to work with logarithms even when they're not readily programmable or easily dividable using typical techniques.