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Algebra II

Find the exact solution, using common logarithms, and a two-decimal-place approximation of each solution,when appropriate.
$5^{x} + 125 (5^{- x}) = 30$
we have to find the exact solution.
$5^{x} + \frac{125}{5 x} = 30$
$5^{x} + \frac{125}{5^{x}} = 30$
let $5^{x} = t$

Henry Winters 2022-04-12

How do I prove that $\sqrt[3]{5 \sqrt{2} + 7} - \sqrt[3]{5 \sqrt{2} - 7}$ is an integer?

Milton Shepard 2022-04-12

For what number $n$ does the equation $\log_{n} x =^{n} x$ have exactly one solution?

Jefferson Pacheco 2022-04-12

Find the following products of complex numbers
$(- 1 - 2 i) (2 + i)$

Colten Welch 2022-04-12

Find the exact solution, using common logarithms, and a two-decimal-place approximation of each solution,when appropriate.
$2^{- x^{2}} = 5$

gabolzm6d 2022-04-12

Find the equation of the tangent to the curve $y = 2^{x} + 2^{- x}$ at the point $(2, 4 \frac{1}{4})$
$y = 2^{x} + 2^{- x}$
$\ln y = x \ln 2 - x \ln 2$
$\ln y = 0$
$\frac{1}{y} \frac{d y}{d x} = 0$
$\frac{d y}{d x} = 0$
I've checked the answer and I've got the differential wrong, What am I doing wrong? I assume it has something to do with the expression with the negative exponent? Am I not allowed to prefix the natural expression with "minus" x? Could someone explain please, thank you!

Rosa Townsend 2022-04-12

Find the distance between the complex numbers in the complex plane.
$- 8 + i$
$- 3 + 8 i$

Ormezzani6cuu 2022-04-12

Find intersection of linear and logarithmic lines
I have equations for two lines, one of which is linear and the other is logarithmic, ie:
$y = m_{1} x + c_{1}$
$y = m_{2} \cdot \ln (x) + c_{2}$
..and I need to find out where (if at all) these lines intersect. I realise that I need to solve:
$m_{1} \cdot x + c_{1} = m_{2} \cdot \ln (x) + c_{2}$
..for x, but apart from shuffling the constants around I'm not sure how to do this
Is there a general solution to this problem?
Thanks

robinmarian9nhn8 2022-04-12

Let $F = C (X, Y, Z)$ be rational function field.
And let $K$ be the splitting field of $F (T) = T^{6} - X T^{4} + Y T^{2} - Z$ .
Please Find all galois extensions $L / F$ such that $L \subset K$ and $[L : F] = 6$ .
I'm stuck on this problem.

vilitatelp014 2022-04-12

Let $C (x, y)$ be a degree 2 extension of $C (x)$ where $y$ is a root of $p (Z) = Z^{2} + (x^{2} + 1)$ .
Is it true that $C (x, y)$ is not a rational function field? In other words, there is no $z \in C (x, y)$ such that $C (x, y) = C (z)$

Alessandra Carrillo 2022-04-11

Use the same-derivative argument, as was done to prove the Productand Power Rules for logarithms, to prove the Quotient Ruleproperty.

Jayleen Aguirre 2022-04-11

Use the properties of logarithms to write each expression as a single term.
a. $\ln [(x + 3) - \ln (x - 1)]$
b. $\frac{1}{2} \log_{7} (2 x - 3) + \log_{7} x - \log_{7} q$

Ean Hughes 2022-04-11

Use the properties of logarithms to solve the following equation for x.
$2 \ln (x + 2) - \ln x = \ln 8$

Leonard Montes 2022-04-11

use the properties of logarithms to rewrite the expression as 3 separate logarithms : $\log_{3} \sqrt[4]{\frac{5}{2}}$

poznateqojh 2022-04-11

Use the properties oflogarithms to simplify the expression.
$3^{x + 1} = 27$

Jazmin Strong 2022-04-11

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places $\log_{π} 400$ .

Adison Palmer 2022-04-11

Unitary matrix in trace and log function
I am trying to do a unitary transformation $U$ on square matrix $A$ which is embedded inside a trace and natural log function, and the following property is supposed to hold:
$t r (\ln (A)) = t r (\ln (U A U^{∔}))$
What property of the $t r$ and $\ln$ functions would allow us to do that?

Jaylyn Villarreal 2022-04-11

Trying to convert a nasty logarithm into an exponential
I have the following equation that I must express in terms of r:
$Δ V = \frac{λ}{2 π ϵ_{0}} \ln (\frac{r}{R})$
This is a pretty tough one. I am not sure how to get the r out of the logarithm. I have tried to express the logarithm as $\ln (r) - \ln (R)$ , , but I am not sure where to go from there, or where the coefficient should go. I know that $Δ V$ goes up in the exponent...

junoonib89p4 2022-04-10

What are the points of discontinuity of the function $\frac{w^{2} - 21 w + 108}{w^{4} + 4 w^{3} - 16 w^{2} - 16 w + 48}$ ?

Leon Robinson 2022-04-10

Compute all the points of discontinuity of $\frac{a^{2} - 25 a + 154}{a^{5} - 10 a^{4} + 21 a^{3} + 28 a^{2} - 52 a - 48}$ .

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