Problem 5 - Contemporary Precalculus

Question

Write the given expression as a single logarithm. $$2(\ln x)-3\left(\ln x^{2}+\ln x\right)$$

Step-by-Step Solution

Answer

Given the expression: \(2(\ln x)-3\left(\ln x^{2}+\ln x\right)\), we can simplify and rewrite it as a single logarithm by following these steps: Step 1: Distribute the coefficients to the logarithms within the parentheses: $$2(\ln x)-3\left(\ln x^{2}+\ln x\right) = 2\ln x - 3\ln x^{2} - 3\ln x$$ Step 2: Apply the power property of logarithms to the expression: $$ 2\ln x - 3\ln x^{2} - 3\ln x = 2\ln x - 6\ln x - 3\ln x$$ Step 3: Combine using the product and quotient properties of logarithms: $$2\ln x - 6\ln x - 3\ln x = \ln x^{2} - \ln x^{6} - \ln x^3 = \ln \left(\frac{x^2}{x^6} \cdot\frac{1}{x^3} \right)$$ Step 4: Simplify the argument of the logarithm: $$\ln \left(\frac{x^2}{x^6} \cdot\frac{1}{x^3}\right) = \ln \left (\frac{x^2}{x^6x^3}\right) = \ln \left (\frac{x^2}{x^9}\right)$$ Step 5: Apply the quotient property of logarithms once more: $$\ln \left (\frac{x^2}{x^9}\right) = \ln x^2 - \ln x^9$$ Therefore, the given expression can be simplified and rewritten as a single logarithm: \(\ln x^2 - \ln x^9\).

Step 1: Simplify the given expression

In the given expression, we need to distribute the coefficients (2 and 3) to the logarithms within the parentheses: $$2(\ln x)-3\left(\ln x^{2}+\ln x\right) = 2\ln x - 3\ln x^{2} - 3\ln x$$

Step 2: Apply the power property

Apply the power property of logarithms to the expression, which states that \(\log a^b=b\log a\): $$ 2\ln x - 3\ln x^{2} - 3\ln x = 2\ln x - 6\ln x - 3\ln x$$

Step 3: Combine using the product and quotient properties

Now, we'll use the product property of logarithms, which states that \(\log a + \log b = \log (a\cdot b)\), and the quotient property, which states that \(\log a - \log b = \log \left(\frac{a}{b}\right)\), to combine the expression into a single logarithm: $$2\ln x - 6\ln x - 3\ln x = \ln x^{2} - \ln x^{6} - \ln x^3 = \ln \left(\frac{x^2}{x^6} \cdot\frac{1}{x^3} \right)$$

Step 4: Simplify the argument

Now, simplify the argument of the logarithm: $$\ln \left(\frac{x^2}{x^6} \cdot\frac{1}{x^3}\right) = \ln \left (\frac{x^2}{x^6x^3}\right) = \ln \left (\frac{x^2}{x^9}\right)$$

Step 5: Apply the quotient property

Finally, apply the quotient property of logarithms once more: $$\ln \left (\frac{x^2}{x^9}\right) = \ln x^2 - \ln x^9$$ Now, we have the given expression written as a single logarithm.

Key Concepts

Logarithmic Properties

Logarithms are mathematical operations that help us deal with exponential relationships. They have unique properties that simplify complex expressions. Understanding these properties can make tasks like multiplication and division of powers much easier.
Logarithms have three main properties which are critical in various calculations:

• Product Property: Helps combine logs with addition.
• Quotient Property: Helps combine logs using subtraction.
• Power Property: Allows you to move the exponent in front of the log.

These properties are essential tools for simplifying expressions with logarithms. They turn complicated multiplications and divisions hidden in exponents into more manageable forms.

Product Property of Logarithms

The product property states that the logarithm of a product can be expressed as the sum of the logarithms of the factors. This property greatly simplifies logarithmic expressions.
Let's use the property:

• If you have ewline \( \log a + \log b = \log (a \cdot b) \)ewline It means you can combine them into a single log by multiplying the bases.

In practical terms, if you're splitting logs due to addition, it's basically finding the log of a product, making chaotic quantities manageable. The product property thus becomes invaluable to compress expressions into simpler forms.

Quotient Property of Logarithms

Quotient property is almost the opposite of the product property. It lets you express the division of two numbers inside a log as the subtraction of two logs. This is useful in reversing the complexity brought by division.
Here’s how it works:

• For ewline \(\log a - \log b = \log \left( \frac{a}{b} \right) \)ewline you subtract two logs to turn it into a single log fractionally representing the division.

The quotient property turns division relationships into single logarithms, capturing the idea of simplifying an intricate division of powers into straightforward subtraction.

Power Property of Logarithms

The power property of logarithms allows us to take any exponent on a logarithmic term and move it in front of the term, as a coefficient. This can substantially simplify the comparison and manipulation of growths and scales.
Use it formidably like this:

• Given ewline \(\log a^b = b \cdot \log a\)ewline the exponent becomes a multiplier in front of the log. This transformation manages exponents effectively and keeps expressions as simple as possible.

Especially when multiple exponential terms are involved, applying the power property diligently helps to unravel complex relationships tied in scaling factors.