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Math`log_4(128)` To evaluate, factor 128. `= log_4 (2^7)` Then, apply the formula of change base `log_b (a) = (log_c (a))/(log_c (b))` . `= (log_2 (2^7))/(log_2 (4))` `= (log_2 (2^7))/(log_2 (2^2))` To...

Math`log_125 (625)` To evaluate, factor 625. `=log_125 (5^4)` Then, apply the formula of change base `log_b(a) = (log_c(a))/(log_c(b))` . `= (log_5 (5^4))/(log_5 (125))` `= (log_5 (5^4))/(log_5...

Math`log_8 (32)` To evaluate, factor 32. `=log_8 (2^5)` Then, apply the formula of change base `log_b(a) = (log_c(a))/(log_c(b))` . `= (log_2 (2^5))/(log_2 (8))` `=(log_2(2^5))/(log_2(2^3))` Also,...

Math`log_27 (9)` To evaluate, factor the 9. `= log_27 (3^2)` Then, apply the formula of change base `log_b (a) =(log_c(a))/(log_c(b))` . `= (log_3(3^2))/(log_3 (27))` `=(log_3 (3^2))/(log_3 (3^3))` To...

MathWe are asked to graph the function `y=log_6(x4)+2 ` : The graph is a translation of the graph of `y=log_6x ` 4 units right and 2 units up. The domain is x>4 and the range is all real numbers....

MathWe are asked to graph the function `y=log_4(x+2)1 ` : This is a translation of the graph of ` y=log_4x ` 2units left and 1 unit down. The domain is x>2 and the range is all real numbers. There...

MathWe are asked to graph the function `y=log_3x+4 ` : The graph is a translation of the graph `y=log_3x ` up 4 units. Some points on the graph: (1/27,1),(1/9,2),(1/3,3),(1,4),(3,5),(9,6) The domain is...

MathWe are asked to graph the function `y=log_2(x3) ` : The graph is a translation of the graph of `y=log_2x ` 3 units right. Some points on the graph:...

Math`2(log_3 (20)  log_3 (4)) + 0.5log_3(4)` First, apply the differencequotient rule of logarithm `log_b (m/n) = log_b(m)  log_b(n)` . `= 2 (log_3 (20/4))+0.5log_3(4)` `=2log_3(5) + 0.5log_3(4)`...

Math`ln40+2ln(1/2) + lnx` First, apply the logarithm rule `log_b (a^m)=m*log_b(a)` . `= ln40 + ln(1/2)^2 + ln x` `=ln40+ ln(1/4) + lnx` And, apply the rule `log_b(m*n) = log_bm +log_b n` . `= ln (40 *...

Math`5log_4(2) + 7log_4(x) + 4log_4(y)` To express this as one logarithm, first apply the rule `log_b a^m = m*log_b(a)` . `= log_4(2^5) + log_4(x^7) + log_4(y^4)` `= log_4(32) + log_4(x^7)+log_4(y^4)`...

MathWe are asked to solve `5^(2x)+20*5^x125=0 ` : Rewrite as ` (5^x)^2+20*5^x125=0 ` and let `y=5^x ` to get ` y^2+20y125=0` and (y+25)(y5)=0 so y=25 or y=5. y cannot be 25 as `5^x>0 ` for all...

MathWe are asked to solve `2^(2x)12*2^x+32=0 ` : Rewrite as `(2^x)^212*2^x+32=0 ` and let `y=2^x ` ; then `y^212y+32=0 ` and (y8)(y4)=0 so y=8 or y=4. y=8 ==> ` 2^x=8 ==> x=3 ` y=4 ==>...

MathTo solve the equation log_3(x)=log_9(6x), we may apply logarithm properties. Apply the logarithm property: `log_a(b)= (log_c(b))/log_c(a)` on `log_3(x)` , we get: `(log_9(x))/(log_9(3))=log_9(6x)`...

MathSolve `log_2(x+1)=log_8(3x) ` : Rewrite using the change of base formula: `(ln(x+1))/(ln(2))=(ln(3x))/(ln(8)) ` `(ln(8))/(ln(2))=(ln(3x))/(ln(x+1)) ` But `ln(8)=ln(2^3)=3ln(2) ` so:...

MathTo solve the equation: `10^(3x8)=2^(5x)` , we may take "ln" on both sides. `ln(10^(3x8))=ln(2^(5x))` Apply natural logarithm property: `ln(x^n) = n*ln(x)` . `(3x8)ln(10)=(5x)ln(2)` Let...

MathRewrite `6^(2x5)' as `(2 times 3)^(2x5) = 3^(2x5)2^(2x5)' Divide both sides of the equation through by `3^(x+4)' giving `1 = 3^(2x5x4)2^(2x5)' That is, `3^(x9)2^(2x5) = 1' Take logs (with...

MathTo evaluate the given equation `log_2(x4)=6` , we may apply the logarithm property: `a^(log_a(x))=x` . Raised both sides by base of `2` . `2^(log_2(x4))=2^6` `x4=64` Add `4` on both sides to...

MathTo evaluate the given equation `1/3log_5(12x)=2` , we may apply logarithm property: `n* log_b(x) = log_b(x^n)` . `log_5((12x)^(1/3))=2` Take the "log" on both sides to be able to apply the...

MathTo evaluate the equation `log_6(3x10)=log_6(145x)` , we apply logarithm property: `a^(log_a(x))=x` . Raised both sides by base of `6` . `6^(log_6(3x10))=6^(log_6(145x))` `3x10=145x` Add `10`...

MathTo solve the equation `log_3(18x+7)=log_3(3x+38)` , we apply logarithm property: `a^(log_a(x))=x` . Raised both sides by base of `3` . `3^(log_3(18x+7))=3^(log_3(3x+38))` `18x+7=3x+38` Subtract 7...

Math`log(12x11)=log(3x+13)` Using the property of logarithmic equality, `12x11=3x+13` `=>12x3x=13+11` `=>9x=24` `=>x=24/9` `=>x=8/3` Let's plug back the solution in the equation to check...

Math`log_5(2x7)=log_5(3x9)` Using one to one property of logarithms, `2x7=3x9` `=>2x3x=9+7` `=>x=2` `=>x=2` Let's plug back the solution in the equation, `log_5(2*27)=log_5(3*29)`...

MathTo evaluate the equation `ln(x+19)=ln(7x8)` , we apply natural logarithm property: `e^(ln(x))=x` . Raise both sides by base of `e` . `e^(ln(x+19))=e^(ln(7x8))` `x+19=7x8` Subtract `7x` from both...

Math`ln(4x7)=ln(x+11)` Using one to one property of logarithms, `4x7=x+11` `=>4xx=11+7` `=>3x=18` `=>x=18/3` `=>x=6` Plug back the solution in the equation to check the solution,...

MathTo solve the equation `log_5(5x+9)=log_5(6x)` , we apply logarithm property: `a^(log_a(x))=x` . Raise both sides by base of `5` . `5^( log_5(5x+9))=5^(log_5(6x))` `5x+9=6x` Subtract `5x` from...

MathTo solve the given equation `7^(6x)=12` , we may take "`ln` " on both sides of the equation. `ln(7^(6x))=ln(12)` Apply natural logarithm property: `n*ln (x)=ln (x^n)` . `6x*ln(7)=ln(12)` Divide...

MathTo solve the given equation `7^(3x)=18` , we may take "ln" on both sides of the equation. `ln(7^(3x))=ln(18)` Apply natural logarithm property:` ln (x^n) = n*ln (x)` . `3xln(7)=ln(18)` Divide both...

MathTo solve the given equation `8^x=20` , we may take "ln" on both sides of the equation. `ln(8^x)=ln(20)` Apply natural logarithm property: `ln (x^n) = n*ln (x)` . `xln(8)=ln(20)` Divide both sides...

MathTo evaluate the given equation `25^(10x+8)=(1/125)^(42x)` , we may apply `25=5^2` and `1/125=5^(3)` . The equation becomes: `(5^2)^(10x+8)=(5^(3))^(42x)` Apply Law of Exponents: `(x^n)^m =...

MathTo evaluate the given equation `10^(3x10)=(1/100)^(6x1)` , we may apply `100=10^2` . The equation becomes: `10^(3x10)=(1/10^2)^(6x1)` Apply Law of Exponents: `1/x^n = x^(n)` ....

MathTo evaluate the given equation `36^(5x+2)=(1/6)^(11x)` , we may apply `36=6^2` and `1/6=6^(1)` . The equation becomes: `(6^2)^(5x+2)=(6^(1))^(11x)` Apply Law of Exponents: `(x^n)^m = x^(n*m)`...

Math`3^(3x7)=81^(123x)` To solve, factor 81. `3^(3x7)=(3^4)^(123x)` To simplify the right side, apply the exponent rule `(a^m)^n=a^(m*n)` . `3^(3x7)=3^(4*(123x))` `3^(3x7)= 3^(4812x)` Since...

MathTo evaluate the given equation `4^(2x5)=64^(3x)` , we may let `64 =4^3` . The equation becomes: `4^(2x5)=(4^3)^(3x)` . Apply Law of exponents: `(x^n)^m = x^(n*m)` . `4^(2x5)=4^(3*3x)`...

Math`27^(4x1)=9^(3x+8)` To solve, factor the 9 and 27. `(3^3)^(4x1)=(3^2)(^(3x+8)` To simplify each side, apply the exponent rule `(a^m)^n=a^(m*n)` . `3^(3*(4x1))=3^(2*(3x+8))`...

Math`8^(x1)=32^(3x2)` To solve, factor 8 and 32. `(2^3)^(x1)=(2^5)^(3x2)` To simplify each side, apply the exponent rule `(a^m)^n = a^(m*n)` . `2^(3*(x1)) = 2^(5*(3x2))` `2^(3x3) = 2^(15x10)`...

Math`7^(3x+4)=49^(2x+1)` To solve, factor the 49. `7^(3x+4)=(7^2)^(2x+1)` To simplify the right side, apply the exponent property `(a^m)^n=a^(m*n)` . `7^(3x+4)=7^(4x+2)` Since the two sides have the...

Math`5^(x4)=25^(x6)` To solve, factor the 25. `5^(x4)=(5^2)^(x6)` To simplify the right side, apply the exponent rule `(a^m)^n = a^(m*n)` . `5^(x4)=5^(2*(x6))` `5^(x4)=5^(2x12)` Since both...

ScienceAn electromagnetic wave propagates in vacuum with the electric field `E_0 cos(kxomega t)` in the...The magnitude of the Poynting vector `S` can be found using the following relation. `S=1/mu_0(E xx B)` `E` ands `B` are perpendicular here so `sin(theta)` in the cross product is one. Therefore,...

ScienceWe will use conservation of energy to solve this problem. We need to consider the rotational energy of the cylinder and the translational energy of the center of mass. `E_i=E_f`...

ReferenceIn the movie, the banks and investment bankers basically collude to enrich themselves at the expense of investors. The main culprit in this fiasco is one Michael Burry, a hedgefund manager (played...

ScienceA square loop of wire of side length `L` containing a load resistor `R` is oriented perpendicular...The magnetic flux through the loop is the magnetic field times the component of the area vector that is parallel to field. `Phi_B=B_0*A cos(theta)=B_0*L^2 cos(omega t)` This generates an...

Law and PoliticsThis article is about which rights are protected under the Due Process Clauses of the 5th and 14th Amendments to the Constitution. The essential question related to this debate is whether these...

HistoryThere are many complexities and even uncertainties surrounding the events that caused the Salem witch trials. Historians do agree on certain facts:  Suspicion about witchcraft had existed for...

LiteratureFirst complete this sentence with appropriate names/words of your choice: "As [name of character]...It sounds like you've been given a creative assignment as a response to reading Franz Kafka's The Metamorphosis! The Metamorphosis is a novella which follows the transformation of Gregor Samsa, a...

LiteratureThis prompt is obviously based on Franz Kafka's story "The Metamorphosis" in which the protagonist, Gregor Samsa, wakes up to discover that he has been transformed into a giant insect. The point of...

ReferenceThe Robber's Cave experiment was conducted in 1954 by a famous social psychologist, Muzafer Sherif. In this experiment, 22 12yearold boys from white, middleclass, Protestant backgrounds with two...

ReferencePrejudice is a negative opinion about a person or group of people that is not based on actual experience. Discrimination occurs when a person acts on their prejudice. The development of prejudice...

PsychologyWhat is meant by shifting standards when judging members of different groups? What is an example?...The concept of shifting standards is defined as the idea that our judgements are based on something we hold to be standard, and these standards may shift depending on multiple factors. For example,...

HistoryBefore we compare and contrast the three systems, let's take a look at what the three represent. The encomienda system was originally instituted by the Spanish monarchy to exact tribute from its...