Question
Rewrite the expression below using the properties of logarithms.
3
log
3
x
+
2
log
2
y
−
4
log
4
z
(1 point)
Responses
log
4
(
x
3
y
2
z
4
)
log sub 4 of open paren the fraction with numerator x cubed y squared and denominator z to the 4th power close paren
log
(
x
3
y
2
z
4
)
the log of open paren the fraction with numerator x cubed y squared and denominator z to the 4th power close paren
log
x
3
+
log
y
2
z
4
log
x
3
+
log
y
2
z
4
log
3
x
3
+
log
2
y
2
−
log
4
z
4
3
log
3
x
+
2
log
2
y
−
4
log
4
z
(1 point)
Responses
log
4
(
x
3
y
2
z
4
)
log sub 4 of open paren the fraction with numerator x cubed y squared and denominator z to the 4th power close paren
log
(
x
3
y
2
z
4
)
the log of open paren the fraction with numerator x cubed y squared and denominator z to the 4th power close paren
log
x
3
+
log
y
2
z
4
log
x
3
+
log
y
2
z
4
log
3
x
3
+
log
2
y
2
−
log
4
z
4
Answers
GPT-4o mini
To rewrite the expression using the properties of logarithms, we start with the given expression:
\[ 3 \log_3 x + 2 \log_2 y - 4 \log_4 z \]
We can use the following properties of logarithms:
1. \( a \log_b c = \log_b (c^a) \) (Power Rule)
2. \( \log_b m + \log_b n = \log_b (mn) \) (Product Rule)
3. \( \log_b m - \log_b n = \log_b \left( \frac{m}{n} \right) \) (Quotient Rule)
Let's apply these properties step by step.
1. **Apply the Power Rule**:
- For \( 3 \log_3 x \):
\[ 3 \log_3 x = \log_3 (x^3) \]
- For \( 2 \log_2 y \):
\[ 2 \log_2 y = \log_2 (y^2) \]
- For \( -4 \log_4 z \):
\[ -4 \log_4 z = -\log_4 (z^4) = \log_4 \left( \frac{1}{z^4} \right) \]
Now rewriting the entire expression:
\[ \log_3 (x^3) + \log_2 (y^2) + \log_4 \left( \frac{1}{z^4} \right) \]
2. **Combine the logs using the Product and Quotient Rules**:
Since the logarithms are with different bases, we cannot combine them directly. However, we can convert them to the same base if necessary, but usually, it would be best to express the combined logarithm in terms of the powers we defined above.
Hence, formally we rewrite the expression as:
\[
\log_3 (x^3) + \log_2 (y^2) + \log_4 \left( \frac{1}{z^4} \right)
\]
This is the simplified form we arrive at without converting to a single logarithm.
Since converting to a different base isn't asked or needed, the final logarithmic expression remains in its distinct logarithmic bases.
So the rewritten expression using the properties of logarithms, closely adhering to your options is best represented with:
**log (x^3) + log (y^2) - log (z^4)** (as a form that highlights the components)
If looking for a single log expression:
- \( \log_4 \left( \frac{x^3 y^2}{z^4} \right) \) is also an answer if converting all to base 4, though requires conversion.
As none of the options perfectly matches the log output expected directly without conversion across logs, your options may require reevaluation depending on context given or a preference on log basis.
\[ 3 \log_3 x + 2 \log_2 y - 4 \log_4 z \]
We can use the following properties of logarithms:
1. \( a \log_b c = \log_b (c^a) \) (Power Rule)
2. \( \log_b m + \log_b n = \log_b (mn) \) (Product Rule)
3. \( \log_b m - \log_b n = \log_b \left( \frac{m}{n} \right) \) (Quotient Rule)
Let's apply these properties step by step.
1. **Apply the Power Rule**:
- For \( 3 \log_3 x \):
\[ 3 \log_3 x = \log_3 (x^3) \]
- For \( 2 \log_2 y \):
\[ 2 \log_2 y = \log_2 (y^2) \]
- For \( -4 \log_4 z \):
\[ -4 \log_4 z = -\log_4 (z^4) = \log_4 \left( \frac{1}{z^4} \right) \]
Now rewriting the entire expression:
\[ \log_3 (x^3) + \log_2 (y^2) + \log_4 \left( \frac{1}{z^4} \right) \]
2. **Combine the logs using the Product and Quotient Rules**:
Since the logarithms are with different bases, we cannot combine them directly. However, we can convert them to the same base if necessary, but usually, it would be best to express the combined logarithm in terms of the powers we defined above.
Hence, formally we rewrite the expression as:
\[
\log_3 (x^3) + \log_2 (y^2) + \log_4 \left( \frac{1}{z^4} \right)
\]
This is the simplified form we arrive at without converting to a single logarithm.
Since converting to a different base isn't asked or needed, the final logarithmic expression remains in its distinct logarithmic bases.
So the rewritten expression using the properties of logarithms, closely adhering to your options is best represented with:
**log (x^3) + log (y^2) - log (z^4)** (as a form that highlights the components)
If looking for a single log expression:
- \( \log_4 \left( \frac{x^3 y^2}{z^4} \right) \) is also an answer if converting all to base 4, though requires conversion.
As none of the options perfectly matches the log output expected directly without conversion across logs, your options may require reevaluation depending on context given or a preference on log basis.