Answer :
To identify the properties used to perform the given logarithmic simplifications, we need to match each simplification with the corresponding logarithmic property.
1. Simplification: [tex]\( \log (100) = \log (400) - \log (4) \)[/tex]
This simplification uses the quotient property of logarithms. The quotient property states that:
[tex]\[ \log_b \left( \frac{a}{c} \right) = \log_b (a) - \log_b (c) \][/tex]
In this case, we can think of [tex]\( \frac{400}{4} \)[/tex] to simplify as:
[tex]\[ \log (100) = \log \left( \frac{400}{4} \right) = \log (400) - \log (4) \][/tex]
2. Simplification: [tex]\( \log \left(x^2\right) = \log (x) + \log (x) \)[/tex]
This simplification uses both the power property of logarithms and the fact that adding the same logarithm twice is equivalent to multiplying that logarithm by 2. The power property states that:
[tex]\[ \log_b (a^c) = c \cdot \log_b (a) \][/tex]
Therefore:
[tex]\[ \log \left(x^2\right) = 2 \cdot \log (x) \][/tex]
Additionally, we recognize:
[tex]\[ 2 \cdot \log (x) = \log (x) + \log (x) \][/tex]
3. Simplification: [tex]\( \log (49) = 2 \log (7) \)[/tex]
This simplification uses the power property of logarithms. The power property states that:
[tex]\[ \log_b (a^c) = c \cdot \log_b (a) \][/tex]
Here we recognize that:
[tex]\[ 49 = 7^2 \][/tex]
Therefore:
[tex]\[ \log (49) = \log (7^2) = 2 \cdot \log (7) \][/tex]
In summary, the properties used in the simplifications are:
1. Quotient property of logarithms.
2. Power property of logarithms (and the arithmetic property of addition).
3. Power property of logarithms.
1. Simplification: [tex]\( \log (100) = \log (400) - \log (4) \)[/tex]
This simplification uses the quotient property of logarithms. The quotient property states that:
[tex]\[ \log_b \left( \frac{a}{c} \right) = \log_b (a) - \log_b (c) \][/tex]
In this case, we can think of [tex]\( \frac{400}{4} \)[/tex] to simplify as:
[tex]\[ \log (100) = \log \left( \frac{400}{4} \right) = \log (400) - \log (4) \][/tex]
2. Simplification: [tex]\( \log \left(x^2\right) = \log (x) + \log (x) \)[/tex]
This simplification uses both the power property of logarithms and the fact that adding the same logarithm twice is equivalent to multiplying that logarithm by 2. The power property states that:
[tex]\[ \log_b (a^c) = c \cdot \log_b (a) \][/tex]
Therefore:
[tex]\[ \log \left(x^2\right) = 2 \cdot \log (x) \][/tex]
Additionally, we recognize:
[tex]\[ 2 \cdot \log (x) = \log (x) + \log (x) \][/tex]
3. Simplification: [tex]\( \log (49) = 2 \log (7) \)[/tex]
This simplification uses the power property of logarithms. The power property states that:
[tex]\[ \log_b (a^c) = c \cdot \log_b (a) \][/tex]
Here we recognize that:
[tex]\[ 49 = 7^2 \][/tex]
Therefore:
[tex]\[ \log (49) = \log (7^2) = 2 \cdot \log (7) \][/tex]
In summary, the properties used in the simplifications are:
1. Quotient property of logarithms.
2. Power property of logarithms (and the arithmetic property of addition).
3. Power property of logarithms.
