Expert Answers to Algebra 2 Problems

To simplify the expression ${\displaystyle \sqrt{45x}+\sqrt{20x}}$.

${\displaystyle \text{If }y=\frac{3^x+2}{3^x-2},\text{ then }x={\log }_3\frac{2y+2}{y+k}\text{ for some real number k. What is the value of k? }}$

Solve the logarithmic equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places.
${\displaystyle {\log }_{a}x+{\log }_a(x-2)={\log }_a(x+4)}$

Consider ${\displaystyle A\left(x\right)=x^3+x^2+1}$ and ${\displaystyle B\left(x\right)=x^2+x+1}$ in ${\displaystyle GF\left(17\right)}$. Also, the irreducible polynomial is given as ${\displaystyle p\left(x\right)=x^3+1}$
1. Find ${\displaystyle A^2\left(x\right)—B\left(x\right)}$. (13 pts)
Hint: Use ${\displaystyle p\left(x\right)}$ if the initial result does not fall in ${\displaystyle GF\left(17\right)}$.
2. Find ${\displaystyle A^2\left(x\right)\cdot B^2\left(x\right)}$. (12 pts)
Hint: Use ${\displaystyle p\left(x\right)}$ if the initial result does not fall in ${\displaystyle GF\left(17\right)}$.

Rewrite the following logarithms in expanded form by applying the properties of logarithms.
${\displaystyle \log \left(x^y\right)}$

For polynomial arithmetic with coefficients in ${\displaystyle Z_{10}}$, perform the following calculations.
a.${\displaystyle (7x+2)-(x^2+5)}$
b.${\displaystyle (6x^2+x+3)\times (5x^2+2)}$

Ise properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator
${\displaystyle {\log }_5\left(\frac{625}{y}\right)}$

Use the properties of logarithms to expand the following expression as much as possible. Simplify any numerical expressions that can be evaluated without a calculator.
${\displaystyle {\log }_4(48x+16y)}$

${\displaystyle \text{The product property of logarithms states that }{\log }_b\left(xy\right)=\text{ for positive real numbers b,x and where }b\ne 1}$

Compound interest Solve the compound interest formula
${\displaystyle A=P{(1+\frac{r}{n})}^{nt}}$
for t by using natural logarithms

${\displaystyle x+\frac{1}{x}=4}$.
Given: ${\displaystyle x=2+\sqrt{3}}$

To simplify the expression ${\displaystyle {25}^{\frac{1}{2}}}$

Use the Properties of Logarithms to condense the logarithm. Simplify if possible. (Assume the variable is positive.)
${\displaystyle {\log }_3(x^2-1)-2{\log }_3(x-1)}$

Use the Laws of Logarithms to evaluate the expression.
${\displaystyle {\log }_3\left(15\right)-{\log }_3\left(40\right)+{\log }_3\left(216\right)}$

To simplify the expression ${\displaystyle {(-27)}^{\frac{1}{3}}}$

The value of ${\displaystyle \sqrt{64}}$.

Explain how you would solve the following equation:
${\displaystyle \ln \left(x\right)+\ln (x-5)=\ln (21-x)}$
Describe why you may only choose some of the possible roots of any polynomial you reduce the problem to
Step 1
By using the property of logarithms, simplify the LHS .
${\displaystyle \ln \left(a\right)+\ln \left(b\right)=\ln \left(ab\right)}$
${\displaystyle \text{Simplify the LHS}}$
${\displaystyle \ln \left(x\right)+\ln (x-5)=\ln \left(x(x-5)\right)}$
${\displaystyle =\ln (x^2-5x)}$
Step 2
Since there is log term on both the sides of the equation, we can take antilog on both sides, simplify and solve the quadratic for x
${\displaystyle \ln (x^2-5x)=\ln (21-x)}$
${\displaystyle e^{\ln (x^2-5x)}=e^{\ln (21-x)}}$
${\displaystyle x^2-5x=x-21}$
${\displaystyle x^2-5x-x-21=0}$
${\displaystyle x^2-4x-21=0}$
${\displaystyle x^2-7x+3x-21=0}$
${\displaystyle x(x-7)+3(x-7)=0}$
${\displaystyle (x-7)(x+3)=0}$
${\displaystyle \Rightarrow x=7,x=-3}$
Step 3
There are two solutions possible for value of x, one positive the other negative. However notice that in the given question there is a term ln(x). But since log of a number is defined only for x>0, we discard x=-3.
Hence the answer is x=7.

Solve the logarithmic equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places.
${\displaystyle 2{\log }_5(x-3)-{\log }_{58}={\log }_{52}}$

Solve for (y) in terms of (x):
${\displaystyle 3\ln y=3\ln x}$

Solve the exponential equation ${\displaystyle e^{4x-5}-7=11243}$. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution.