Chapter 10: Q. 423 (page 1065)

In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
$\log_{5} \frac{8 a^{2} b^{6} c}{d^{3}}$ .

Short Answer

Expert verified

The solution is $3 \log_{5} (2) + 2 \log_{5} (a) + 6 \log_{5} (b) + \log_{5} (c) - 3 \log_{5} (d)$ .

Step by step solution

Step 1. Given and explanation.

We have $\log_{5} \frac{8 a^{2} b^{6} c}{d^{3}}$ .

The product property states that if there is an expression as $\log_{a} (m \cdot n)$ , it is equal to $\log_{a} m + \log_{a} n$ .

The quotient property states that if there is an expression as $\log_{a} \frac{m}{n}$ , it is equal to $\log_{a} m - \log_{a} n$ .

Applying these properties, we get the solution.

Step 2. Expanding using quotient property.

We have $\log_{5} \frac{8 a^{2} b^{6} c}{d^{3}}$ .

Using the quotient property, we can write this as ,

$\log_{5} (8 a^{2} b^{6} c) - \log_{5} (d^{3})$ .

Step 3. Expanding using product property.

We have $\log_{5} (8 a^{2} b^{6} c) - \log_{5} (d^{3})$ .

Using the product rule, we get
$= \log_{5} (8 a^{2} b^{6} c) - \log_{5} (d^{3}) = [(\log_{5} 8) + (\log_{5} a^{2}) + (\log_{5} b^{6}) + (\log_{5} c)] - \log_{5} (d^{3}) = \log_{5} {(2)}^{3} + \log_{5} {(a)}^{2} + \log_{5} {(b)}^{6} + \log_{5} c - \log_{5} {(d)}^{3} = 3 \log_{5} (2) + 2 \log_{5} (a) + 6 \log_{5} (b) + \log_{5} (c) - 3 \log_{5} (d)$
Thus we get the value.

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In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
$\log_{5} \frac{8 a^{2} b^{6} c}{d^{3}}$ .

Short Answer

Step by step solution

Step 1. Given and explanation.

Step 2. Expanding using quotient property.

Step 3. Expanding using product property.

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In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.log58a2b6cd3.

Short Answer

Step by step solution

Step 1. Given and explanation.

Step 2. Expanding using quotient property.

Step 3. Expanding using product property.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

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Study anywhere. Anytime. Across all devices.

In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
$\log_{5} \frac{8 a^{2} b^{6} c}{d^{3}}$ .