Question

Rewrite the expression below using the properties of logarithms. 3log3x+2log2y−4log4z (1 point) Responses log3x3+log2y2−log4z4 log sub 3 x cubed plus log sub 2 y squared minus log sub 4 z to the 4th power logx3+logy2z4 logx3+logy2z4 log4(x3y2z4) log sub 4 of open paren the fraction with numerator x cubed y squared and denominator z to the 4th power close paren log(x3y2z4)

Answer 1

To rewrite the expression \( 3\log_3 x + 2\log_2 y - 4\log_4 z \) using the properties of logarithms, we can apply the power rule and the change of base formula.

1. **Apply the power rule:**
- The power rule states that \( n \log_b a = \log_b a^n \).
- So:
- \( 3\log_3 x = \log_3 x^3 \)
- \( 2\log_2 y = \log_2 y^2 \)
- \( -4\log_4 z = -\log_4 z^4 \) (this is equivalent to \( \log_4 z^{-4} \))

2. **Combine the logarithmic expressions:**
- Since we have a subtraction, we can combine the last term into the logarithm using the properties of logs:
\[
\log_3 x^3 + \log_2 y^2 - \log_4 z^4 = \log_3 x^3 + \log_2 y^2 + \log_4(z^{-4})
\]

3. Now, since the bases of the logarithms are different, we can’t directly combine them into a single logarithm unless we switch to a common base.

However, if the final intended output is to express this in a consolidated logarithmic form, we may have:
\[
\log_4\left(\frac{x^3 y^2}{z^4}\right)
\]
This means we combine \( x^3 \) and \( y^2 \) in the numerator dividing by \( z^4 \).

Thus the final answer is:
\[
\log_4\left(\frac{x^3 y^2}{z^4}\right)
\]