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Use the quotient property of logarithms to write the logarithm as a difference of logarithms. Then simplify if possible. Assume that the variables represent positive real numbers.

[tex]\[ \log_{15}\left(\frac{x}{y}\right) = \log_{15}(x) - \log_{15}(y) \][/tex]

Sagot :

GinnyAnsweridn
Sure! Let's solve the given logarithmic expression step-by-step using the properties of logarithms.

We start with the given expression:

[tex]\[ \log_{15}\left(\frac{x}{y}\right) \][/tex]

### Step 1: Use the Quotient Property of Logarithms

One of the key properties of logarithms is the quotient property, which states:

[tex]\[ \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \][/tex]

Here, we can identify [tex]\( m = x \)[/tex] and [tex]\( n = y \)[/tex], and apply this property to our expression:

[tex]\[ \log_{15}\left(\frac{x}{y}\right) = \log_{15}(x) - \log_{15}(y) \][/tex]

### Step 2: Verify Simplification

We have:

[tex]\[ \log_{15}(x) - \log_{15}(y) \][/tex]

In this case, since no specific values for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are provided, there is no further simplification possible from this point.

So, the simplified form of the given logarithmic expression using the quotient property of logarithms is:

[tex]\[ \log_{15}\left(\frac{x}{y}\right) = \log_{15}(x) - \log_{15}(y) \][/tex]

This is the final answer.
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