Problem 42

Question

$\operatorname{In} 36-47,$ if $\log 3=x$ and $\log 5=y,$ write each of the logs in terms of $x$ and $y$ $$ \log \frac{1}{3} $$

Step-by-Step Solution

Verified

Answer

$ \log \frac{1}{3} = -x $.

1Step 1: Understanding the Logarithmic Expression

We are given that $ \log 3 = x $ and $ \log 5 = y $. We need to express $ \log \frac{1}{3} $ in terms of $ x $ and $ y $. To do this, recognize that $ \log \frac{a}{b} = \log a - \log b $. Also, note $ \log \frac{1}{3} = \log 1 - \log 3 $. Since $ \log 1 = 0 $, the expression simplifies to $ 0 - \log 3 $.

2Step 2: Applying Logarithmic Properties

Since we established that $ \log \frac{1}{3} = 0 - \log 3 $, we simplify this to $ -\log 3 $.

3Step 3: Substitute Given Values

Given $ \log 3 = x $, substitute $ x $ for $ \log 3 $ in the expression. This gives us $ -x $.

4Step 4: Final Expression

Now, $ \log \frac{1}{3} $ is expressed in terms of $ x $ as $ -x $.

Key Concepts

Logarithm PropertiesChange of Base FormulaNegative Logarithms

Logarithm Properties

Logarithms have a variety of properties that make them very useful in simplifying complex problems. One useful property is the logarithm of a quotient, which states that the logarithm of the division of two numbers is the difference between the logarithms of those numbers:

$ \log \frac{a}{b} = \log a - \log b $

Additionally, it's important to remember that the logarithm of 1 is always 0, due to the fact that any number raised to the power of 0 equals 1:

$ \log 1 = 0 $

These properties are crucial when breaking down more complex expressions into simpler parts. For example, to simplify $ \log \frac{1}{3} $, we rewrite it as $ \log 1 - \log 3 $. Since $ \log 1 = 0 $, it simplifies directly to $-\log 3$.

Change of Base Formula

The change of base formula is an important tool when working with logarithms of different bases. It allows you to express a logarithm in one base in terms of logarithms in another base:

$ \log_b a = \frac{\log_c a}{\log_c b} $

This is particularly useful when you want to convert logarithms so they can be calculated using a calculator that may only accept logs of certain bases, such as base 10 or base $e$. Although the exercise at hand directly involves base 10 logarithms, it's good to keep this formula in mind. This way, you can always relate different base logarithms to base 10 or natural logarithms, making computations a breeze.

Negative Logarithms

Negative logarithms occur naturally when dealing with numbers less than 1, especially when one of the numbers is the denominator. For instance, consider the expression $ \log \frac{1}{3} $. When simplified, this becomes $-\log 3$. Why does this happen?

Using the property $ \log \frac{a}{b} = \log a - \log b $, we have $ \log 1 - \log 3 $.
Since $ \log 1 = 0 $ as any number to the power of zero is 1, you are left with just $-\log 3$.

This negative sign indicates that the number 3 is in the denominator, reflecting the ‘smallness’ of the fraction $ \frac{1}{3} $. Whenever you find a negative logarithm, it's often pointing towards a fraction that is less than one. It’s a helpful concept to remember, especially when solving problems involving decimals or fractions.

Problem 42

Other exercises in this chapter

Problem 41

Write each expression in terms of $A$ and $B$ if $\log _{2} x=A$ and $\log _{2} y=B$. $\log _{2}\left(x^{2} \div y^{3}\right)$

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Problem 42

In $33-44,$ if $\ln 2=x$ and $\ln 3=y,$ write each of the natural logs in terms of $x$ and $y$ $$ \ln \frac{1}{36} $$

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Problem 42

In $27-56,$ evaluate each logarithmic expression. Show all work. $$ \log _{5} 125 $$

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Problem 42

Write each expression in terms of $A$ and $B$ if $\log _{2} x=A$ and $\log _{2} y=B$. $\log _{2} \sqrt{x y}$

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