Question

Write each expression in terms of individual logarithms of \(x, y,\) and \(z\). a) \(\log _{7} x y^{2} \sqrt{z}\) b) \(\log _{5}(x y z)^{8}\) c) \(\log \frac{x^{2}}{y \sqrt[3]{z}}\) d) \(\log _{3} x \sqrt{\frac{y}{z}}\)

Short answer

a) \(\log _{7} x + 2\log _{7} y + \frac{1}{2} \log _{7} z\) b) \(8(\log _{5} x + \log _{5} y + \log _{5} z)\) c) \(2 \log x - \log y - \frac{1}{3} \log z\) d) \(\log _{3} x + \frac{1}{2} \log _{3} y - \frac{1}{2} \log _{3} z\)

Step-by-step solution

Title - Expand logarithm using product rule

For expression a) \[\log _{7} x y^{2} \sqrt{z} = \log _{7} (x) + \log _{7} (y^{2}) + \log _{7} (\sqrt{z})\]

Title - Apply power rule to each term

Using the power rule of logarithms, \[\log _{7} x y^{2} \sqrt{z} = \log _{7} x + 2\log _{7} y + \frac{1}{2} \log _{7} z\]

Title - Expand using power and product rules

For expression b) \[\log _{5}(x y z)^{8} = 8 \log _{5}(x y z)\]

Title - Apply product rule inside the logarithm

Using the product rule of logarithms, \[8 \log _{5}(x y z) = 8(\log _{5}(x) + \log _{5}(y) + \log _{5}(z))\]

Title - Expand using quotient and power rules

For expression c) \[\log \frac{x^{2}}{y \sqrt[3]{z}} = \log (x^{2}) - \log (y \sqrt[3]{z})\]

Title - Apply product rule inside the logarithm

Using the product rule of logarithms, \[\log (y \sqrt[3]{z}) = \log (y) + \log (\sqrt[3]{z})\]

Title - Apply power rule to each term

Using the power rule of logarithms, \[\log x^{2} - (\log y + \frac{1}{3} \log z) = 2 \log x - \log y - \frac{1}{3} \log z\]

Title - Expand using quotient and power rules

For expression d) \[\log _{3} x \sqrt{\frac{y}{z}} = \log _{3} x + \log _{3} \sqrt{\frac{y}{z}}\]

Title - Apply the power and quotient rules

Using the power rule and applying them to the roots, \[= \log _{3} x + \frac{1}{2} \log _{3} \frac{y}{z} = \log _{3} x + \frac{1}{2}(\log _{3} y - \log _{3} z) = \log _{3} x + \frac{1}{2} \log _{3} y - \frac{1}{2} \log _{3} z\]

Key concepts

Logarithm Properties

Understanding logarithm properties is essential for expanding and simplifying logarithmic expressions. These properties help you manipulate logarithms to make calculations simpler.

Here are the main properties to remember:

• The **Product Rule**: \(\text{log}_b (MN) = \text{log}_b (M) + \text{log}_b (N)\)
• The **Quotient Rule**: \(\text{log}_b \frac{M}{N} = \text{log}_b (M) - \text{log}_b (N)\)
• The **Power Rule**: \(\text{log}_b (M^k) = k \text{log}_b (M)\)

Each of these rules makes it easier to break down complex logarithmic expressions into simpler terms. These properties act as tools that let you rewrite logarithms in a more manageable form.

Product Rule in Logarithms

The product rule in logarithms allows us to split the logarithm of a product into a sum of individual logarithms.

Example:
Consider \(\text{log}_7 (xy^2\text{√}z)\). Applying the product rule, we get:

\(\text{log}_7 (xy^2\text{√}z) = \text{log}_7 (x) + \text{log}_7 (y^2) + \text{log}_7 (\text{√}z)\)

This makes it easier to apply further rules like the power rule.

Breaking down an expression using the product rule helps to simplify calculations.

Power Rule in Logarithms

The power rule in logarithms is used to move an exponent out in front of the logarithm. This can be very useful for simplifying logarithmic expressions.

Example:
For \(\text{log}_7 (y^2)\), we can apply the power rule:

\(\text{log}_7 (y^2) = 2 \text{log}_7 (y)\)

Similarly:

\(\text{log}_7 (\text{√}z) = \text{log}_7 (z^{1/2}) = \frac{1}{2}\text{log}_7 (z)\)

Applying the power rule makes it easier to interact with the terms individually.

Quotient Rule in Logarithms

The quotient rule lets you split the logarithm of a fraction into the difference between the logarithms of the numerator and the denominator.

Example:
Consider \(\text{log} \frac{x^2}{y \text{√³}z}\). Using the quotient rule, we get:

\(\text{log} \frac{x^2}{y \text{√³}z} = \text{log} (x^2) - \text{log} (y \text{√³}z)\)

We can then apply the product rule and power rule within these terms for further simplification:

\(\text{log} (x^2) - (\text{log} (y) + \text{log} (\text{√³}z)) = 2\text{log}(x) - \text{log}(y) - \frac{1}{3} \text{log}(z)\)

This process will help in breaking down the complex fraction into simpler parts easier to work with.