Use the properties of logarithms to evaluate the expression.

$$\log _{3} 54-\log _{3} 2$$

Let S be the Cartesian coordinate plane $\mathbb{R} \times \mathbb{R}$ and define a relation R on S by (a,b) R (c,d) iff a = c. Verify that R is an equivalence relation and describe a typical equivalence class $E_{(a, b)}$.

Find the partial sum.

$$\sum_{k=1}^{4}\left(\frac{5}{3}\right)^{k-1}$$

Pie Bakery owns 60 percent of Slice Products Company's stock. During 20X8, Slice produced 100,000 bags of flour, which it sold to Pie Bakery for $900,000. On December 31, 20X8, Pie had 20,000 bags of flour purchased from Slice Products on hand. Slice prices its sales at cost plus 50 percent of cost for profit. Pie, which purchased all its flour from Slice in 20X8, had no inventory on hand on January 1, 20X8.

Pie Bakery reported income from its baking operations of$400,000, and Slice Products reported net income of $150,000 for 20X8.

Required

Compute the amount reported as cost of goods sold in the 20X8 consolidated income statement.

Write the expression $\frac{\log 90}{4 \log 5}, \quad b=25$ in the form $\log _b x$ for the given value of $b$. State the value of $x$, and verify your answer using a calculator.

Use a change-of-base formula to evaluate each logarithm.

(a) $\log _2 17$ $\qquad$(b) $\log _5(0.78)$

Suppose that $S=\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is linearly independent and $\mathbf{w}_{1}=\mathbf{v}_{2} \quad \mathbf{w}_{2}=\mathbf{v}_{1}+\mathbf{v}_{3}$ and $\mathbf{w}_{3}=\mathbf{v}_{1}+\mathbf{v}_{2}+\mathbf{v}_{3}$. Determine whether the set $T=\left\{\mathbf{w}_{\mathbf{1}}, \mathbf{w}_{\mathbf{2}}, \mathbf{w}_{3}\right\}$ is linearly independent or linearly dependent.

In the following problem, is the statement true or false?

$$\log \sqrt{x}=\frac{1}{2} \log x$$

Use the properties of logarithms to condense the expression.

$$5 \log _{3} x+\log _{3}(x-6)$$

Tú y otro(a) estudiante preparan una fiesta de cumpleaños. Pregúntale lo que debes hacer.Sigue el modelo.(Ask a partner whatyou should do to preparefor a party. Follow the model.)

modelo: comprar el papel

$\hspace{31pt}$de regalo

A. ¿Debo comprar el papel de regalo?

B. Sí, cómpralo.

limpiar la cocina

The value of a stock over the past five years where $\mathrm{x}$ is the number of years is modeled by $x^3-2 x^2+5 x-10$. Factor this expression.

Five eighths of a pound is how many ounces?

Solve the system of linear equations by substitution. Check your answer.

$$\left\{\begin{array}{c}x+2 y=6 \\ 2 x+4 y=12\end{array}\right.$$

STEP 1 Solve $x+2 y=6$ for $x$.

$$x= \square$$

STEP 2 Substitute that expression for $x$ into $2 x+4 y=12$.

$$\begin{aligned} 2( \square)+ 4y &=12 \\ \square- \square y+ \square y &= 12 \\ \square &= 12 \end{aligned}$$

Solution: ________

Evaluate the expression for the exercise given below without using a calculator.

$$\log 25+\log 4$$