Problem 60 - Elementary and Intermediate Algebra

Question

Write logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. \(\log 4 x z^{2}\)

Step-by-Step Solution

Answer

\(\log 4 + \log x + 2\log z\)

Step 1: Apply the Product Rule for Logarithms

Begin by using the product rule for logarithms, which states that the logarithm of a product is the sum of the logarithms of its factors. Apply this rule to separate the expression into different logarithms. Thus, \( \log(4xz^2) = \log 4 + \log x + \log(z^2) \).

Step 2: Apply the Power Rule for Logarithms

Next, apply the power rule for logarithms to simplify \( \log(z^2) \). The power rule states that \( \log(a^b) = b\log a \). Using this rule, rewrite \( \log(z^2) \) as \( 2\log z \). The expression then becomes \( \log 4 + \log x + 2\log z \).

Step 3: Conclusion

Combine the results to express the original logarithm as a sum of simpler logarithms. The expression can now be rewritten as \( \log 4 + \log x + 2\log z \). This is the simplified form of \( \log 4xz^2 \).

Key Concepts

Product Rule for Logarithms

When you're dealing with logarithms, one of the most useful tools is the product rule. This rule is your go-to method for splitting up logarithms whenever you see a product inside. It tells us that the logarithm of a product is the sum of the logarithms of each individual factor.

In mathematical terms, the product rule is expressed as:

• If you have a product inside a logarithm like \( \log(a \times b) \), you can break it down to \( \log a + \log b \).

Let's apply this to our exercise, \( \log 4xz^2 \). The logarithm of this product can be split into:\( \log 4 + \log x + \log(z^2) \). Each part comes from one of the factors: 4, \( x \), and \( z^2 \). You just add up their individual logs to get the new expression. This makes complex expressions much simpler to handle and understand.

Power Rule for Logarithms

Once you break down a logarithmic expression using the product rule, the power rule for logarithms might come in handy next. This rule simplifies logarithms that have exponents, making it easier to manage and understand them.

The power rule is quite straightforward:

• For an expression like \( \log(a^b) \), you can take the exponent \( b \) and move it out front, resulting in \( b \log a \).

Now, by applying the power rule to \( \log(z^2) \) in our expression, you get:\( 2\log z \). This allows the expression to be simplified from \( \log(z^2) \) into something more straightforward. Your initial expression \( \log 4 + \log x + \log(z^2) \) turns into \( \log 4 + \log x + 2\log z \).

This demonstration shows how reducing exponents in logarithms can simplify otherwise cumbersome expressions.

Simplifying Logarithmic Expressions

Once you've applied the product and power rules, the last step is ensuring the expression is in its most simplified form. Simplification makes the expression easier to interpret and use in further calculations.

In our example, the expression \( \log 4 + \log x + 2\log z \) is now in a neat and simplified form. It efficiently breaks down the original, more complicated term of \( \log 4xz^2 \). Once you pull apart the logs into a sum and simplify any exponents using the power rule, you're left with individual logarithmic terms.

To simplify logarithmic expressions, remember:

• Ensure each term is expressed as the logarithm of a single quantity.
• Apply rules like the product and power rules carefully, to ensure no steps are skipped.

This approach helps you tackle any logarithmic expression with confidence, by turning it into a series of simpler, easier-to-manage parts.