# Expand Logarithms Using The Quotient Property Worksheet

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Expanding logarithms using the quotient property involves breaking down a single logarithm of a quotient into the difference of logarithms of its numerator and denominator. For example, \log_b\left(\frac{x}{y}\right) expands to $$\log_b(x) - \log_b(y)$$ when the bases are the same. This way of expanding logarithms makes complex math problems easier by breaking them into simpler parts, which helps with doing math faster and more accurately.
Example: Expand the logarithm \log_3\left(\frac{10}{x}\right) using the quotient property.

Algebra 2
Logarithms

## How Will This Worksheet on "Expand Logarithms Using the Quotient Property" Benefit Your Student's Learning?

• When we break down logarithms using the quotient property, we are making complex math problems easier to handle by splitting them into simpler parts.
• Transforming divisions into subtractions of logarithms helps perform calculations more quickly because we are dealing with fewer separate components.
• Understanding how to expand logarithms using the quotient property helps us solve harder math problems more effectively.
• Learning how to work with logarithmic expressions prepares us for advanced math like calculus and algebra by giving us a strong base.
• Breaking down complicated logarithmic terms into simpler ones helps us avoid mistakes in our math, making sure our answers are right.

## How to Expand Logarithms Using the Quotient Property?

• Start with a logarithm of a quotient, such as \log_b\left(\frac{x}{y}\right).
• Use the quotient property of logarithms, which states \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y), ensuring the bases are the same.
• Separate the logarithm into the difference of logarithms of the numerator and the denominator.
• Expand the logarithmic expression by writing it as a difference of simpler logarithmic terms, making it easier to work with in calculations and problem-solving.

## Solved Example

Q. Expand the logarithm. Assume all expressions exist and are well-defined. Write your answer as a sum or difference of common logarithms or multiples of common logarithms. The inside of each logarithm must be a distinct constant or variable. $\log \frac{y}{z}$
Solution:
1. Identify Property: Identify the property used to expand $\log \left( \frac{y}{z} \right)$. $\newline$Quotient property: $\log_b \left( \frac{P}{Q} \right) = \log_b P - \log_b Q$
2. Apply Quotient Property: Apply the quotient property to expand $\log \left( \frac{y}{z} \right)$. $\newline$$\log \left( \frac{y}{z} \right) = \log(y) - \log(z)$

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