Problem 6 - Contemporary Precalculus

Question

Write the given expression as a single logarithm. $$-\log \left(\frac{3 \sqrt{x}}{2}\right)-\log (\sqrt{5 x})$$

Step-by-Step Solution

Answer

Question: Rewrite the expression as a single logarithm: \(-\log \left(\frac{3 \sqrt{x}}{2}\right)-\log (\sqrt{5 x})\) Answer: \(\log \left(\frac{1}{\frac{15x}{2}}\right)\)

Step 1: Apply the power rule to given expression

$$-\log \left(\frac{3 \sqrt{x}}{2}\right)-\log (\sqrt{5 x}) =\ -\log \left(\frac{3 x^{\frac{1}{2}}}{2}\right)-\log (5x^{\frac{1}{2}})$$ Step 2: Apply the quotient rule Now that the expressions have been rewritten with exponents, we can apply the quotient rule. The quotient rule states \(\log_a(x)-\log_a(y)=\log_a(\frac{x}{y})\). We will combine both terms under one logarithm.

Step 2: Apply the quotient rule to the expression

$$-\log \left(\frac{3 x^{\frac{1}{2}}}{2}\right)-\log (5x^{\frac{1}{2}}) = \log \left(\frac{1}{\frac{3 x^{\frac{1}{2}}}{2} \cdot 5x^{\frac{1}{2}}}\right)$$ Step 3: Simplify the expression Lastly, we need to simplify the expression inside the logarithm.

Step 3: Simplify the expression

$$\log \left(\frac{1}{\frac{3 x^{\frac{1}{2}}}{2} \cdot 5x^{\frac{1}{2}}}\right) = \log \left(\frac{1}{\frac{15x}{2}}\right)$$ Now, we can rewrite the expression as a single logarithm.

Step 4: Write the given expression as a single logarithm

$$-\log \left(\frac{3 \sqrt{x}}{2}\right)-\log (\sqrt{5 x}) = \log \left(\frac{1}{\frac{15x}{2}}\right)$$

Key Concepts

logarithm properties

Logarithms are powerful mathematical tools that can simplify complex expressions and solve equations involving exponentials. They have properties that allow us to manipulate and combine them in useful ways. Some essential logarithm properties include:

• Product Rule: States that the logarithm of a product is the sum of the logarithms. For example, \(\log_b(xy) = \log_b(x) + \log_b(y)\).
• Quotient Rule: Indicates that the logarithm of a quotient is the difference of the logarithms. This can be expressed as \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\).
• Power Rule: Describes that the logarithm of an exponential expression can be simplified by bringing the exponent in front, \(\log_b(x^k) = k \cdot \log_b(x)\).

Understanding these properties helps in manipulating and simplifying logarithmic expressions effectively. In our example, applying these rules allows us to combine two separate logarithmic terms into a single log expression.

simplifying expressions

Simplifying expressions is a vital skill in solving mathematical problems. When dealing with logarithms, it involves reducing the terms into their simplest form so they are easier to interpret or compute. Look at the original expression:
\(-\log \left(\frac{3 \sqrt{x}}{2}\right)-\log (\sqrt{5 x})\). Here’s how we simplify it step-by-step:

• Rewrite Roots as Exponents: Instead of writing roots, use their equivalent exponential form. Remember that \(\sqrt{x} = x^{1/2}\). So, \(\sqrt{3x} = 3x^{1/2}\).
• Apply the Power Rule: Since any coefficient in the argument can be considered part of the exponent, it can be multiplied with the logarithm, streamlining the expression.
  

Simplification can always make evaluating or computing expressions more straightforward when the final calculation or interpretation is needed.

combining logarithms

Combining logarithms involves using various logarithmic properties to consolidate multiple log terms into one single expression. Starting with our original task, which was to combine:
\(-\log \left(\frac{3 \sqrt{x}}{2}\right) - \log (\sqrt{5 x})\). Our goal is to combine them into a unified expression:

• Use the Quotient Rule for Logarithms: This rule helps in combining the separate logarithm terms into a single expression by condensing the respective arguments into a quotient.
• Combine into a Single Logarithm: Integrating the terms using the quotient rule results in \(\log \left(\frac{1}{{\frac{3 x^{1/2}}{2}} \cdot \sqrt{5x}}\right)\).

Combining logarithms by simplifying their structure and retaining accuracy is essential for problem-solving. It allows us to work with a single term rather than multiple different ones, making the math less cumbersome.