Problem 16 - Algebra 2

Question

Write each logarithmic expression as a single logarithm. \(\log 5-k \log 2\)

Step-by-Step Solution

Answer

\( \log \frac{5}{2^k} \)

Step 1: Apply the power rule of logarithms

First, apply the power rule on \( k \log 2 \). This rule allows us to rewrite \( k \log 2 \) as \( \log 2^k \). So the expression now reads \( \log 5 - \log 2^k \).

Step 2: Apply the quotient rule of logarithms

Then, apply the quotient rule, which states that the difference of two logs can be rewritten as a quotient within a single logarithm. Applying this to the current expression will result to the following equation: \( \log \frac{5}{2^k} \). This is now in the form of a single logarithm.

Key Concepts

Power Rule

The power rule is a handy trick when dealing with logarithms and exponents. It tells us how to work with expressions like \(k \log b\). The power rule states:

• \(k \log b = \log b^k\)

This means that if you have a constant \(k\) multiplied by a log, you can bring \(k\) upstairs as an exponent to \(b\) inside the log.
This makes it much easier to apply other rules later on in an equation.
For example, in the expression \(k \log 2\), using the power rule, we can rewrite it as \(\log 2^k\).
This doesn't change the value--it just prepares it better for further simplification.
Once simplified, the expression becomes more manageable for further operations like using the quotient rule.

Quotient Rule

The quotient rule is an essential tool in simplifying logarithmic expressions involving subtraction.
When you see a subtraction between two logarithms like \(\log a - \log b\), you can use the quotient rule:

• \(\log a - \log b = \log \frac{a}{b}\)

This rule essentially lets us turn a subtraction of logarithms into the log of a division.
This makes it simpler to handle and reduces the complexity of the expression.
For example, after applying the power rule on the expression \(\log 5 - \log 2^k\), the quotient rule allows us to condense it further into a single term: \(\log \frac{5}{2^k}\).
By restructuring the expression this way, it becomes easier to read and calculate.

Logarithmic Expression

Logarithmic expressions are composed of logarithms, which indicate the power we need to raise a base number to get a certain value.
In log expressions, we use rules like the power and quotient rule to simplify them.
The goal is to make complex logs easier to manage, especially when they appear in larger equations.
For instance, an expression like \(\log 5 - k \log 2\) might initially seem challenging.

• By using the power rule, it turns into \(\log 5 - \log 2^k\).
• Then, using the quotient rule, we further simplify it to a single logarithm: \(\log \frac{5}{2^k}\).

This simplification is beneficial because single log expressions are often much simpler to work with or integrate into larger mathematical frameworks.

Mathematical Expression Simplification

Simplifying mathematical expressions is crucial when solving problems, especially with logs.
The process involves applying rules like the power and quotient rules to reduce expressions to their most basic form.
Simplified expressions are easier to interpret and solve, reducing the possibility of errors.

• When faced with a complex log expression, start by identifying which rules can help.
• Apply these rules step-by-step to rewrite the expression into a simpler form.

For the expression \(\log 5 - k \log 2\), simplifying it involved using the power rule to change it to \(\log 5 - \log 2^k\).
Then, the quotient rule made it a single log \(\log \frac{5}{2^k}\).
This step-by-step simplification reveals the underlying structure of a problem, making it easier to understand and solve.