Problem 130 - College Algebra Essentials

Question

If \(\log 3=A\) and \(\log 7=B,\) find \(\log _{7} 9\) in terms of \(A\) and \(B\).

Step-by-Step Solution

Answer

\(\log _{7} 9 = \frac{2A}{B}\)

Step 1 Property of logarithms

The property of logarithms states that \(\log _{b} a= \frac{1}{\log _{a} b}\). This is the principle that should be utilized to solve this exercise.

Step 2 Apply the property of logarithms

Applying this property to the required \(\log _{7} 9\), it gets converted into \(\frac{1}{\log _{9} 7}\).

Step 3 Convert log base 9

The base 9 can be expressed as \(3^{2}\). So \(\log_{9}7\) can be expressed as \(\frac{\log_{3}7}{\log_{3}9}\), or after simplifying fraction, \(\frac{\log_{3}7}{2}\).

Step 4 Substitute given log values

We now substitute given values: \(\log 3=A\) and \(\log 7=B\) into \(\frac{\log_{3}7}{2}\). This gives us \(\frac{B}{2A}\).

Step 5 Substitute into base conversion formula

Finally substitute \(\frac{B}{2A}\) into the formula in step 2, we get \(\log _{7} 9 = \frac{2A}{B}\).

Key Concepts

Logarithmic Properties

Logarithmic properties are essential tools that help in simplifying and solving logarithmic expressions. These include various rules like the product, quotient, and power rules. Another key property used in this exercise is the reciprocal property:

• For two bases, if \(log_b a = x\), then \(log_a b = \frac{1}{x}\).

This property allows us to switch the base and the argument. It is useful because it transforms the base of the logarithm, making it easier to match with provided information. Understanding these properties provides a foundation for tackling a broad range of logarithmic problems. They allow you to rewrite expressions in forms that reveal hidden relationships.

Base Conversion

Base conversion is about changing the base of a logarithmic expression to another one. This is particularly helpful when you have logarithmic values like \(\log 3 = A\) and \(\log 7 = B\) linked to a logarithmic equation with a different base.

• In this exercise, we need to convert \(\log_{7} 9\) into terms of known values.
• By using the property \(log_b a = \frac{1}{log_a b}\), we change the base 7 to base 9.

Once converted, further manipulation simplifies the expression into terms of known logs. Converting bases requires applying these properties correctly, ensuring any values provided can directly substitute into the equation.

Logarithmic Equations

Logarithmic equations involve solving for unknowns within logarithms. Understanding how logs correlate and converting these into simpler forms enable solutions. In the exercise, \(\log_{7} 9\) is resolved through transformation:

• By breaking down into simpler components using known values \(\log 3 = A\) and \(\log 7 = B\).
• Expressing complex logs like \(\log_9 7\) by known logs develops into \(\frac{B}{2A}\).

Logarithmic equations can feel complex, but with properties and known log values, they simplify predictably. Solving them eventually comes down to understanding how to manage and rearrange logarithmic forms effectively.

Change of Base Formula

The change of base formula is a tool that converts logarithms from one base to another, typically used when calculators or known values are unavailable for a specific base. The formula is expressed as:

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

This formula means any logarithm can be rewritten with a different base, making flexibility in solving problems viable.
In our exercise, the base needed conversion for \(\log_{9} 7\) into a base aligned with known values: \(\frac{log_3 7}{2 log_3 3}\). Such flexibility allows the use of familiar values (\(\log 3 = A\) and \(\log 7 = B\)) in unfamiliar base situations. Understanding and mastering the change of base formula opens doors to solving diverse logarithmic equations.