Question

1. reasoning which of the following is not equivalent to (log_{b}\frac{y^{4}}{3x})? justify your answer.

a (4log_{b}y - log_{b}3x)
b (4log_{b}y - log_{b}3 - log_{b}x)
c (4log_{b}y - log_{b}3 + log_{b}x)
d (log_{b}y^{4} - log_{b}3 - log_{b}x)

1. reasoning which of the following equations is correct? justify your answer.

a (log_{b}x + 2log_{b}y = log_{b}(x + y^{2}))
b (9log_{b}x - 2log_{b}y = log_{b}\frac{x^{9}}{y^{2}})
c (5log_{b}x + 7log_{b}y = log_{b}x^{5}y^{7})
d (log_{b}x - 5log_{b}y = log_{b}\frac{x}{5y})

section 6.5 properties of logarithms 331

Explanation:

Problem 31

Step1: Simplify the target expression

Use quotient rule: $\log_b\frac{m}{n}=\log_b m-\log_b n$
$\log_3 \frac{y^4}{3x} = \log_3 y^4 - \log_3 (3x)$

Step2: Expand with power/product rules

Power rule: $\log_b m^n=n\log_b m$; Product rule: $\log_b(mn)=\log_b m+\log_b n$
$\log_3 y^4 - \log_3 (3x) = 4\log_3 y - (\log_3 3 + \log_3 x)$

Step3: Simplify $\log_3 3$

$\log_3 3 = 1$, so:
$4\log_3 y - 1 - \log_3 x = 4\log_3 y - \log_3 x - 1$

Step4: Compare with options

Option A: $4\log_3 y - \log_3 3x = 4\log_3 y - (\log_3 3+\log_3 x)=4\log_3 y -1-\log_3 x$ (equivalent)
Option B: $4\log_3 y - \log_3 3 + \log_3 x = 4\log_3 y -1+\log_3 x$ (not equivalent)
Option C: $4\log_3 y - \log_3 3 - \log_3 x = 4\log_3 y -1-\log_3 x$ (equivalent)
Option D: $\log_3 y^4 - \log_3 3 - \log_3 x = 4\log_3 y -1-\log_3 x$ (equivalent)

Step1: Test Option A

Sum rule: $\log_b m + \log_b n=\log_b(mn)$, but $\log_x y + 2\log_x y = 3\log_x y = \log_x y^3
eq \log_x(x + y^2)$ (incorrect)

Step2: Test Option B

Difference rule: $\log_b m - \log_b n=\log_b\frac{m}{n}$; Power rule: $n\log_b m=\log_b m^n$
$9\log x - 2\log y = \log x^9 - \log y^2 = \log \frac{x^9}{y^2}
eq \log \frac{x^9}{y}$ (incorrect)

Step3: Test Option C

Power rule: $n\log_b m=\log_b m^n$; Sum rule: $\log_b m + \log_b n=\log_b(mn)$
$5\log_x x + 7\log_x y = \log_x x^5 + \log_x y^7 = \log_x(x^5 y^7)$ (correct)

Step4: Test Option D

Difference rule: $\log_b m - \log_b n=\log_b\frac{m}{n}$
$5\log x - 5\log y = 5(\log x - \log y) = \log \frac{x^5}{y^5}
eq \log \frac{x}{5y}$ (incorrect)

Answer:

B. $4 \log_3 y - \log_3 3 + \log_3 x$

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Problem 32