
Math
I assume that you are asking about results that either lie outside a confidence interval, or results during a hypothesis test that lie in the critical region (tail.) When creating a confidence...

Math
This is an excellent question to ask, and it happens to be one of those questions where the answer is actually very clear. Statistics wins. Whether you see yourself in a career based in the...

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...

Math
We are asked to graph the following function: `y=log_5(x+1)3` The base function is `y=log_5(x)` and the graph will be a translation of 1 unit left and 3 units down of the graph of the base...

Math
We 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....

Math
We 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...

Math
We 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...

Math
We 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
We are asked to graph the following function: `y=log_(1/5)(x)` The domain is x>0 and the range is all real numbers. The graph is decreasing and concave up on the domain. Some points on the...

Math
We are asked to graph the following function: `y=log_(1/3)(x)` The domain is x>0 and the range is all real numbers. The graph of the function is decreasing and concave up on its domain. Some...

Math
We are asked to graph the following function: `y=log_6(x)` The domain is x>0 and the range is all real numbers. The graph is increasing on the domain, and the graph is concave down on the...

Math
We are asked to graph the following function: `y=log_4(x)` The domain is x>0 and the range is all real numbers. The graph increases on its domain and is concave down on its domain. Some points...

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)`...

Math
We 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...

Math
We 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 ==>...

Math
To 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:...

Math
Solve `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:...

Math
To 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...

Math
`3^(x+4) = 6^(2x5)` To solve, take the natural logarithm of both sides. `ln (3^(x+4)) = ln (6^(2x5))` To simplify each side, apply the logarithm rule `ln (a^m) =m*ln(a)` . `(x+4)ln(3) = (2x5) ln...

Math
To evaluate the given equation `log_6(3x)+log_6(x1)=3` , we may apply the logarithm property: `log_b(x)+log_b(y)=log_b(x*y)` . `log_6(3x)+log_6(x1)=3` `log_6(3x*(x1))=3` `log_6(3x^23x)=3` To...

Math
We are asked to solve the following equation: `4ln(x)+3=21` Use basic algebraic rules to isolate the term with the logarithm: `4ln(x)=18` `ln(x)=9/2` Exponentiate both sides with base e:...

Math
To evaluate the given equation `log_4(x)+log_4(x+10)=2` , we may apply the logarithm property: `log_b(x)+log_b(y)=log_b(x*y)` . `log_4(x)+log_4(x+10)=2` `log_4((x)*(x+10))=2`...

Math
To 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...

Math
To 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...

Math
We are asked to solve `log_8(512x)=log_8(6x1)` Exponentiating both sides with base 8 we get: 512x=6x118x=6 x=1/3. This is in the domain of both expressions of the equality (the domain for the...

Math
To 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`...

Math
To 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)`...

Math
To 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,...

Math
To 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...

Math
For the given equation `2^(0.1x)5=12` , we may simplify by combining like terms. Add `5` on both sides of the equation. `2^(0.1x)5+5=12+5` `2^(0.1x)=17` Take the "`ln` " on both sides to be able...

Math
For the given equation `0.5^x0.25=4` , we may simplify by combining like terms. Add `0.25` on both sides of the equation. `0.5^x0.25+0. 25=4+0.25` `0.5^x=4.25` Take the "`ln` " on both sides to...

Math
For the given equation `10^(3x)+4 =9` , we may simplify by combining like terms. Subtract 4 from both sides of the equation. `10^(3x)+44 =94` `10^(3x)=5` Take the "ln" on both sides to be able to...

Math
To 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...

Math
To 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...

Math
To 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...

Math
To 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 =...

Math
To 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)` ....

Math
To 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...

Math
To 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))` `3^(12x3)=3^(6x+16)`...

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)`...