Question

Write each expression as a single logarithm in simplest form. State any restrictions on the variable. a) \(\log _{5} x+\log _{5} \sqrt{x^{3}}-2 \log _{5} x\) b) \(\log _{11} \frac{x}{\sqrt{x}}+\log _{11} \sqrt{x^{5}}-\frac{7}{3} \log _{11} x\)

Short answer

a) \ \ \( \log_5 \sqrt{x} \), x > 0 \ \ \ \ b) \ \( \log_{11} (x^{2/3}) \), x > 0

Step-by-step solution

Title - Combine Logarithmic Terms

Use the properties of logarithms to combine the given expression. Recall that for any logarithms with the same base, \ \ \( \log_b x + \log_b y = \log_b (xy) \) and \ \ \( \log_b x - \log_b y = \log_b (\frac{x}{y}) \).

Title - Simplify Each Part a)

\[ \log_5 x + \log_5 \sqrt{x^3} - 2 \log_5 x \] \ \ Combine the first two logarithms: \ \ \( \log_5 x + \log_5 \sqrt{x^3} = \log_5 (x \cdot \sqrt{x^3}) \) \ \ We get: \ \ \( \log_5 (x \cdot x^{3/2}) = \log_5 (x^{5/2}) \) \ \ Now subtract the third term: \ \ \( \log_5 (x^{5/2}) - 2 \log_5 x = \log_5 (x^{5/2} \cdot x^{-2}) \) \ \ \( = \log_5 (x^{1/2}) = \log_5 \sqrt{x} \).

Title - State Restrictions for a)

For the expression \ \ \( \log_5 \sqrt{x} \), x must be greater than zero. Thus, \ \ \( x > 0 \).

Title - Simplify Each Part b)

\[ \log_{11} \frac{x}{\sqrt{x}} + \log_{11} \sqrt{x^5} - \frac{7}{3} \log_{11} x \] \ \ Combine the first two logarithms using product property: \ \ \( \log_{11} \frac{x}{\sqrt{x}} = \log_{11} (x \cdot x^{-1/2}) = \log_{11} (x^{1/2}) \) \ \ Combine: \ \ \( \log_{11} (x^{1/2}) + \log_{11} (x^{5/2}) \) \ \ We get: \ \ \( \log_{11} (x^{1/2} \cdot x^{5/2}) = \log_{11} (x^{3}) \) \ \ Now subtract the third term: \ \ \( \log_{11} (x^{3}) - \frac{7}{3} \log_{11} x = \log_{11} (x^{3} \cdot x^{-7/3}) \) \ \ \( = \log_{11} (x^{9/3 - 7/3}) = \log_{11} (x^{2/3}) \).

Title - State Restrictions for b)

For the expression \ \ \( \log_{11} (x^{2/3}) \), x must be greater than zero. Thus, \ \ \( x > 0 \).

Key concepts

Properties of Logarithms

Understanding the properties of logarithms is essential for simplifying logarithmic expressions. Here are some key properties to remember:

- The **Product Property**: \( \log_b (xy) = \log_b x + \log_b y \) allows you to combine two logs into one when you are adding logs of the same base.
- The **Quotient Property**: \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \) helps you combine two logs into one when you are subtracting logs of the same base.
- The **Power Property**: \( \log_b \left( x^k \right) = k \log_b x \) allows you to move a power in a logarithmic expression to the front as a multiplier.

These properties are used to combine and simplify logarithmic expressions step by step.
For instance, in the given exercise, we use these properties to simplify and combine multiple logarithm terms into one. Understanding how and when to apply these properties can significantly simplify complex logarithmic expressions. Remember, these properties only work if the logarithms have the same base.

Logarithmic Expressions

Logarithmic expressions can often seem complex, but by breaking them down using the properties of logarithms, they become much more manageable.

Let's consider the expression from part (a) of the exercise:
\[ \log_5 x + \log_5 \sqrt{x^3} - 2 \log_5 x \]

We first recognize that all logarithms have the same base (5 in this case). By using the Product Property, we combine \log_5 x and \log_5 \sqrt{x^3} into one logarithm:
\[ \log_5 (x \cdot x^{3/2}) ewline = \log_5 (x^{1} \cdot x^{3/2}) ewline = \log_5 (x^{5/2}) \]

Next, we apply the Quotient Property to subtract \log_5 x:
\log_5 (x^{5/2}) - 2 \log_5 x ewline = \log_5 (x^{5/2} \cdot x^{-2}) ewline = \log_5 (x^{1/2}) = \log_5 \sqrt{x}

This shows that by using these properties step-by-step, we can simplify the expression significantly.
Similarly, breaking down logarithmic expressions using properties can turn lengthy and complex-looking equations into simpler forms.

Variable Restrictions in Logarithms

When dealing with logarithmic expressions, it is crucial to consider restrictions on the variables involved to ensure the math is valid.

Logarithms are only defined for positive real numbers. This means any variable inside a logarithmic function must be greater than zero.
If we look at the solution for part (a) of the exercise:
\[ \log_5 \sqrt{x} ewline x > 0 \]

This restriction comes from the definition of the logarithm. Since \sqrt{x}\ needs x to be positive (you cannot take the square root of a negative number in the real number system), it directly translates to \log_5 \sqrt{x} having x > 0.

For example, in part (b) with the expression:
\[ \log_{11} (x^{2/3}) \]

to be defined, x must also be greater than zero (x > 0).
Understanding these restrictions ensures your logarithmic evaluations remain valid and prevent undefined log operations. Always check the domain of your expressions to comply with these restrictions.