Problem 25 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{6}\left(\frac{36}{\sqrt{x+1}}\right) $$

Step-by-Step Solution

Answer

The simplified logarithmic expression is \(2 - \frac{1}{2}\log_6{(x+1)}\).

Step 1: Apply the Quotient Rule of Logarithms

According to the quotient rule, \(\log_b(M/N)=\log_bM - \log_bN\). Applying this to given expression: \(\log _{6}\left(\frac{36}{\sqrt{x+1}}\right)=\log_6{36} - \log_6\sqrt{x+1}\).

Step 2: Evaluating \(\log_636\)

The log base 6 of 36 is 2; because \(6^2 = 36\). Then the equation becomes \(2 - \log_6\sqrt{x+1}\).

Step 3: Write Square Root Term as Power Half Term

The square root \(\sqrt{x+1}\) can be rewritten as \((x+1)^{1/2}\). Replace this in the expression to get \(2 - \log_6{(x+1)^{1/2}}\).

Step 4: Apply the Power Rule of Logarithms

According to the power rule, \(\log_bM^p=p\log_bM\). Apply this rule to obtain \(2 - \frac{1}{2}*\log_6{(x+1)}\).

Step 5: Simplify the Expression

Multiplying 1/2 with the logarithm, we get the fully expanded form for the original expression: \(2 - \frac{1}{2}\log_6{(x+1)}\)

Key Concepts

Quotient Rule of Logarithms

When working with logarithms, the Quotient Rule is quite handy and allows you to simplify logarithmic expressions that involve division. The rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator.
For example,

• Given \(\log_b\left(\frac{M}{N}\right)\), this can be rewritten as \(\log_bM - \log_bN\).

In our original problem expression, \(\log _{6}\left(\frac{36}{\sqrt{x+1}}\right)\), we identified \(M=36\) and \(N=\sqrt{x+1}\).
Apply the rule to separate them into two simpler logarithmic terms: \(\log_6{36} - \log_6\sqrt{x+1}\).
This simplification makes it easier to work with each part individually.

Power Rule of Logarithms

The Power Rule is a splendid tool for simplifying logarithmic expressions with exponents. It states that the logarithm of a number raised to an exponent is the exponent times the logarithm of that number.
Here's the general form:

• For \(\log_bM^p\), it becomes \(p\log_bM\).

In our specific case, after evaluating \(\log_636\) and transforming \(\sqrt{x+1}\) into \(\left(x+1\right)^{1/2}\), the Power Rule allows us to simplify the expression further.
By applying it, \(\log_6\left(x+1\right)^{1/2}\) turns into \(\frac{1}{2}\log_6{\left(x+1\right)}\).
This method transforms our expression into a more manageable form, merging powers and logarithms efficiently.

Logarithmic Expressions

Logarithmic expressions consist of logarithms which are the inverse operations of exponentials. They are used to calculate the power to which a base number must be raised to produce a given number.
Understanding the properties of logarithmic expressions is crucial, as these allow for power, product, and quotient manipulations.
In the expression \(\log_6\left(\frac{36}{\sqrt{x+1}}\right)\), you need to break it down using known properties.
Evaluate each part:

• Start with constant base numbers, like \(\log_636\).
• Continue by manipulating more complex components, such as \(\sqrt{x+1}\).

This understanding helps in expanding or condensing expressions, an essential skill in algebra.

Expanding Logarithms

Expanding logarithms is the process of breaking down a compound logarithmic expression into simpler parts.
This approach is not only useful in simplifying results but also in solving complex problems.

• In our problem, we expanded \(\log_6\left(\frac{36}{\sqrt{x+1}}\right)\) by applying the Quotient Rule first, transforming it into \(\log_636 - \log_6\sqrt{x+1}\).
• The Power Rule was then used to further expand \(\log_6\sqrt{x+1}\) into \(\frac{1}{2}\log_6{(x+1)}\).

These steps are essential for complete analysis.
Expanding allows you to handle each part separately, making it easier to compute or evaluate the result. As a result, you gain a clearer picture of the expression's structure and complexity.