Problem 59 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Apply the Product Rule for Logarithms

The product rule for logarithms states that the log of a product is the sum of the logs of the factors. For our expression, we can use this rule to break down the product inside the logarithm:\[\log (xyz^2) = \log x + \log y + \log z^2\]

Step 2: Apply the Power Rule for Logarithms

The power rule for logarithms states that the log of a power can be written as the exponent times the log of the base. For the term \(\log z^2\), apply the power rule:\[\log z^2 = 2 \log z\]Substitute back into the expression:\[\log x + \log y + 2 \log z\]

Step 3: Simplify

The expression \(\log x + \log y + 2 \log z\) is now simplified and written as a sum of logarithms. Each term is in its simplest form, so no further simplification is needed.

Key Concepts

Product Rule for Logarithms

The product rule for logarithms is a fundamental identity that helps simplify logarithmic expressions involving products. When you have the logarithm of a product, like \(\log (xyz^2)\), you can use this rule to split the expression into the sum of separate logarithms.

The product rule essentially states that:

• \( \log(a \times b) = \log a + \log b \)

So, when given \(\log(xyz^2)\), applying the product rule lets us express it as:

• \( \log x + \log y + \log z^2 \)

This breakdown makes it easier to manage and simplifies the calculation of the logarithmic expression, especially when you have multiple variables in a product.

Power Rule for Logarithms

The power rule for logarithms is another useful identity that allows you to handle expressions involving exponents efficiently. When one of the factors in your logarithmic expression is raised to a power, you use this rule to simplify the logarithm of that power.

The power rule states:

• \( \log(a^b) = b \cdot \log a \)

Applying the power rule to the expression \(\log z^2\) from our previous breakdown, we get:

• \( \log z^2 = 2 \cdot \log z \)

The exponent 2 is brought in front, multiplying the logarithm of the base. This results in the expression \(\log x + \log y + 2 \cdot \log z\), which is easier to work with and see clearly influences of each variable and their powers.

Simplifying Logarithmic Expressions

After applying both the product and power rules, the original expression of \(\log xyz^2\) has been transformed into a simpler sum of separate logarithms: \(\log x + \log y + 2 \cdot \log z\). Simplifying logarithmic expressions means rewriting them in a way that's easier to understand and work with, particularly in advanced mathematical operations.

Key steps in simplification are:

• Breaking down composite expressions into simpler parts using logarithmic identities.
• Using the power rule to deal with exponents within the logarithm.
• Combining like terms if possible, though in this particular case, no further combination is required, as each term is already simplified.

Through simplification, it becomes more manageable to substitute specific values if needed or further manipulate the expression in broader mathematical contexts.