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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:...

MathWe 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...

MathWe 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...

MathWe 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...

MathWe 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)`...

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:...

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...

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...

MathTo 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...

MathWe 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:...

MathTo 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`...

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...

MathWe 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...

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...

MathFor 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...

MathFor 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...

MathFor 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...

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))` `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)`...

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...

MathTo determine the power function `y=ax^b` from the given coordinates: `(5,10)` and `(12,81)` , we setup system of equations by plugin the values of x and y on `y=ax^b.` Using the coordinate...

MathTo determine the power function `y=ax^b` from the given coordinates: `(4,8) ` and `(8,30)` , we setup system of equations by plugin the values of `x` and `y` on `y=ax^b` . Using the coordinate...

MathWe are asked to write a power function whose graph includes the points (3,14) and (9,44): `14=a3^b,44=a9^b` From the first equation we get: `a=14/(3^b)` Then `44=(14/(3^b))*9^b` `44=14*3^b`...

MathWe are asked to write a power function whose graph includes the points (2,3) and (6,12). Substitute the given x,y values into the base equation to get two equations in the two unknowns a,b. Solve...

MathWe are asked to write a power function whose graph includes the points (5,9) and (8,34). Substitute the given x,y pairs into the base model to get two equations with the two unknowns a,b. Solve the...

MathWe are asked to write the equation for a power function whose graph passes through the points (4,3) and (8,15). We substitute the known values of x and y into the basic equation to get two...