Problem 4 - Contemporary Precalculus

Question

Write the given expression as a single logarithm. $$3(\log 2 x)-4[\log x-\log (y-5)]$$

Step-by-Step Solution

Answer

Question: Simplify the expression \(3(\log 2x) - 4(\log x - \log (y-5))\) and write it as a single logarithm. Answer: \(\log \frac{(2x)^3(y-5)^4}{x^4}\)

Step 1: Distribute the constants

Distribute 3 and -4 to the respective logarithmic terms: $$3(\log 2x) - 4(\log x - \log (y-5))$$ $$= 3\log 2x - 4\log x + 4\log (y-5)$$

Step 2: Apply the power rule

The power rule states that \(a\log_b(x) = \log_b(x^a)\). Apply this rule to the coefficients of the logarithms: $$= \log (2x)^3 - \log x^4 + \log (y-5)^4$$

Step 3: Apply the product rule

The product rule states that \(\log_b(x) + \log_b(y) = \log_b(xy)\). Combine the first and last logarithms: $$= [\log (2x)^3 + \log (y-5)^4] - \log x^4$$

Step 4: Apply the product rule again

Apply the product rule to the expression in the brackets: $$= \log [(2x)^3(y-5)^4] - \log x^4$$

Step 5: Apply the quotient rule

The quotient rule states that \(\log_b(x) - \log_b(y) = \log_b(\frac{x}{y})\). Apply this rule to combine the logarithms: $$= \log \frac{(2x)^3(y-5)^4}{x^4}$$ Now, the given expression has been simplified and written as a single logarithm.

Key Concepts

Power Rule

The power rule of logarithms is a fundamental tool that helps simplify expressions by handling coefficients in front of log terms. It states that if you have a logarithm of a number raised to an exponent, like \(a \log_b(x)\), it can be rewritten as \(\log_b(x^a)\). This means you essentially "move" the coefficient as an exponent inside the log.

For example: If you have the term \(3 \log 2x\), applying the power rule changes it to \(\log((2x)^3)\). Similarly, for \(-4 \log x\), it becomes \(\log(x^{-4})\). It's a helpful rule for transforming and ultimately simplifying expressions that involve multiple logs.

Product Rule

The product rule of logarithms allows us to combine two separate logarithmic terms into one. It states that \(\log_b(x) + \log_b(y) = \log_b(xy)\). When you see two logs being added, think of this rule. It combines these separate logs into a single term that is much easier to handle.

The product rule was first applied in our expression between \(\log (2x)^3\) and \(\log (y-5)^4\). Their combination using the product rule gives us \(\log((2x)^3 (y-5)^4)\). By doing this, we have successfully merged two parts of the expression, greatly simplifying the overall problem. This step is crucial in moving toward writing the expression as a single logarithm.

Quotient Rule

The quotient rule is another vital principle in logarithms, helping to manage subtraction between logs. It states that \(\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)\). This rule is used when you have a situation where two logs are being subtracted.

In the provided solution, after using the product rule, we reached \([\log((2x)^3(y-5)^4)] - \log(x^4)\). Applying the quotient rule here simplifies it further to \(\log\left(\frac{(2x)^3(y-5)^4}{x^4}\right)\). Thus, the quotient rule helps to turn a subtraction of two logs into a division under a single log, streamlining the expression and bringing it closer to its simplest form.

Simplifying Expressions

Simplifying logarithmic expressions often involves a strategic application of rules such as the power, product, and quotient rules. By systematically applying these rules, complex expressions can be transformed into much simpler ones.

Let's take what we just learned from the exercise. We began with a multi-part expression and modified it using the power rule to adjust exponents, then used the product rule to combine logs that were added, followed by the quotient rule to manage logs that were subtracted. Each step strategically reduced complexity, turning a long and potentially confusing expression into a streamlined single logarithm: \(\log\left(\frac{(2x)^3(y-5)^4}{x^4}\right)\).

These methods show not only how to manipulate log expressions effectively but also teach the flexibility and creativity involved in algebraic transformations. With practice, these steps become intuitive, allowing easier handling of complex logarithmic challenges.