
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)`...

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...

Math
To 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...

Math
To 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...

Math
We 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`...

Math
We 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...

Math
We 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...

Math
We 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...

Math
The given two points of the exponential function are (1,40) and (3,640). To determine the exponential function `y=ab^x` plugin the given x and y values. For the first point (1,40), plugin x=1...

Math
The given two points of the exponential function are (1,2) and (3,50). To determine the exponential function `y=ab^x` plugin the given x and y values. For the first point (1,2), plugin x=1 and...

Math
The given two points of the exponential function are (3,27) and (5,243). To determine the exponential function `y=ab^x` plugin the given x and y values. For the first point (3,27), the values of x...

Math
To determine the power function `y=ax^b` from the given coordinates: `(3,1)` and `(5,4)` , we setup system of equations by plugin the values of x and y on `y=ax^b` . Using the coordinate `(3,1)`...

Math
The given two points of the exponential function are (2,24) and (3,144). To determine the exponential function `y=ab^x` plugin the given x and y values. For the first point (2,24), the values of x...

Math
The given two points of the exponential function are (1,3) and (2,12). To determine the exponential function `y=ab^x` plugin the given x and y values. For the first point (1,3), plugin x = 1 and...

Math
We are asked to write the equation of the parabola with directrix x=1/18 and vertex at the origin: The equation for a parabola with vertex at the origin and focus (a,0) is `y^2=4ax ` Since the...

Math
We are asked to write the equation of the parabola with directrix y=5/12 and vertex at the origin: The equation for a parabola with vertex at the origin and focus (0,a) is `x^2=4ay ` Since the...

Math
We are asked to write the equation of the parabola with vertex at the origin and directrix x=11: The equation of a parabola with vertex at the origin and focus at (a,0) is `y^2=4ax ` ; the parabola...

Math
A parabola with directrix at` x=a` implies that the parabola may opens up sideways towards to the left or right. The position of the directrix with respect to the vertex point can be used to...

Math
A parabola with directrix at `y=k` implies that the parabola may opens up towards upward or downward direction. The position of the directrix with respect to the vertex point can be used to...

Math
A parabola with directrix at `y=k` implies that the parabola may open up towards upward or downward direction. The position of the directrix with respect to the vertex point can be used to...