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MathThe formula for the circumference of a circle is pi*d, where pi is a constant (pi approximately 3.1415926) and d is the diameter of the circle. Another formula for the circumference is C=2*pi*r...

MathFirst of all, with any series problem, we must first decide if it is an arithmetic series or a geometric series. An arithmetic series is one that you can add or subtract to get from the first...

MathThe value of the function `f(x) = (x^2  x  2)/(x  2)` is defined for all values of x other than x = 2. If we try to determine the value of f(x) at x = 2, we get `lim_(x>2) (x^2  x  2)/(x ...

MathAs per fundamental theorem of calculus, `F(t)=int_a^bf(t)dt=F(b)F(a)` where F(t) is the antiderivative of f(t) and F'(t)=f(t), `d/dxint_a^bf(t)dt=b'F'(b)a'F'(a)` Here a=0 so F(a)=0 , b=x^5...

MathFirst make a shorthand for the different types of articles to make the working clearer: p  no. of periodicals P n  no. of novels N w  no. of newspapers W (because N is already taken) h  no....

MathHello! The velocity is the derivative of the displacement function. Therefore we can find the displacement as the function of time by integrating the velocity function. For v(t) =` t^3  11t^2 +...

Mathxe^(x) + e^(x) = 0 Take the `e^(x) ` as common ,we get => `e^(x) [1x]=0` => ` ((1x)/(e^x)) =0` => `1x =0` =>` x= 1`

Mathe^(2x)  2xe^(2x) = 0 Taking the term e^(2x) as common we get, => e^(2x)[12x]=0 => ((12x)/(e^(2x))) =0 => (12x) = 0 * (e^(2x)) => 12x =0 => x = 1/2 = 0.500 the below...

MathGiven 2xln(x) + x = 0 => 2x ln(x) = x cancelling x on both sides we get => 2 ln(x) = 1 => ln(x) = (1/2) => x = e^(1/2) = 0.6065 the graph of the equations 2x ln(x) +x =0 and y= 2x...

Math`1lnx=0 ` `lnx=1 ` Change to `log_e ` `log_(e)x=1 ` `e^1=x ` ` ` `x=2.719 `

Math`1+lnx=0 ` `lnx=1 ` Change to `log_(e) ` `log_(e)x=1 ` `e^1=x ` `x=0.368 `

MathGiven `2xln(1/x)  x=0` `x(2ln(1/x) 1)=0` so now, => `x= 0 or 2ln(1/x)  1 =0` =>`2ln(1/x)  1 =0` => `2ln(1/x) =1` => `ln(1/x) = 1/2` => `1/x = e^(1/2)` => `x = e^(1/2) =...

Math`lnx=7 ` Change to `log_e ` `log_(e)x=7 ` `e^7=x ` `x=1096.633 `

MathChange to `log_e ` `log_(e)6x=2.1 ` `e^2.1=6x ` `x=e^2.1/6 ` `x=1.361 `

MathAssume `log_10 ` `log_(10)3z=2 ` `10^2=3z ` `z=100/3 ` `z=33.333 `

Math`ln5x=10/3 ` Change to `log_e ` `log_(e)5x=10/3 ` `e^(10/3)=5x ` `x=(e^(10/3))/5 ` `x=5.606 `

MathChange to `log_e ` `log_(e)sqrt(x8)=5 ` `e^5=sqrt(x8) ` `(e^(5))^(2)=x8` `e^10+8=x ` `x=22034.466 `

Math`6lnx=8 ` `lnx=4/3 ` Change to `log_e ` `log_(e)x=4/3 ` `e^(4/3)=x ` `x=0.264 `

Math`3lnx=10 ` `lnx=10/3 ` Change to `log_e ` `log_(e)x=10/3 ` `e^(10/3)=x ` `x=28.032 `

Math`log_(3)0.5x=11/6 ` `3^(11/6)=0.5x ` Simplify, `x=14.988 `

MathAssume `log_10 ` `log_(10)x6=11/4 ` `10^(11/4)=x6 ` `10^(11/4)+6=x ` Simplify, `x=568.341 `

MathUse the properties of logarithms to condense the ln's `ln(x/(x+1))=2 ` Rewrite with log `log_(e)(x/(x+1))=2 ` `(x/(x+1))=e^2 ` `(x+1)/x=1/e^(2) ` `1+1/x=1/e^(2) ` `1/x=1/(e^(2))1 `...

MathUse properties of logarithms for the ln's `lnx(x+1)=1 ` `e=x(x+1) ` `e=x^2+x ` `x^2+xe=0 ` Use the quadratic formula `x=1.223 ` and `x=2.223 ` However, since you cannot take the log or ln of any...

MathUse the properties of logarithms to condense the left side of the equation `ln(x+5)=ln((x1)/(x+1)) ` Since you have 2 ln's, set the inner parts equal to each other `x+5=(x1)/(x+1) ` Simplify,...

MathCondense the equation using properties of logs `ln((x+1)/(x2))=lnx ` Since there are ln's on both sides, set the inner parts equal to each other `(x+1)/(x2)=x` Cross multiply and simplify,...

MathSince you have logs on both sides, just set the inner parts equal to each other `3x+4=x10 ` From here, it is simple algebra, just simplify, `x=7 ` But this is not considered a real solution since...

MathSince they are both `log_2 ` , rewrite it using properties of logs `log_(2)(x)(x+2)=log_2(x+6) ` Exponentiate both sides by 2 and solve for x `2^(log_(2)(x)(x+2))= 2^(log_2(x+6))` `x(x+2)=x+6 `...

MathNote that `log_4(4)=1 ` `log_4xlog_4(x1)=(1/2)log_4(4) ` Use log rules and condense `log_(4)(x/(x1))=log_4(4^(1/2)) ` Exponentiate both sides by 4 and cancel logs: `x/(x1)=sqrt4 ` `(x1)/x=1/2...

MathGiven `log(8x)  log(1 + sqrt(x)) = 2` (1) On simplification we get => As we know `log(a)  log(b) = log(a/b)` ` ` so , => ` log(8x)  log(1 + sqrt(x)) = 2` ` ` =>...

MathGiven `2(x^2)e^(2x) + 2xe^(2x) = 0` => `(e^(2x)) *(2(x^2) +2x)=0` =>`2x(e^(2x)) *(x +1) =0` =>` 2x(e^(2x)) = 0 or (x+1) =0` => as `(e^(2x))` cannot be zero so, `x = 0 ` and in `(x+1) =0...

MathGiven (x^2)e^(x) + 2xe^(x) = 0 => e^(x) [2x  x^2]=0 => (2x  x^2)/(e^(x) )=0 as (e^(x) ) cannot be zero so 2x x^2 = 0 => x(2x) =0 => x= 0 or x= 2 the graphs of the equations...

MathDivide by 8 `3^(6x)=5 ` Take the ln of both sides and bring down the exponent `(6x)ln3=ln5 ` Simplify, `x=ln5/ln3+6 ` `x=4.535 `

MathTake the ln of both sides `lne^(3x)=ln12 ` Bring down the exponent `3xlne=ln12 ` Simplify, `x=0.828 `

MathDivide both sides by 1000 `e^(4x)=3/40 ` Take the ln of both sides and bring down the exponent `4xlne=ln(3/40) ` Simplify, `x=0.648 `

MathSubtract 7 from both sides `2e^x=2 ` Divide both sides by 2 `e^x=1 ` Take the ln of both sides and bring down the exponent `xlne = ln1 ` Simplify, x=0

MathAdd 14 to both sides `3e^x=25 ` Divide by 3 `e^x=25/3 ` Take the ln of both sides and bring down the exponent `xlne=ln(25/3) ` Simplify, `x=2.120 `

MathAdd 7 to both sides `6(2^(3x1))=16 ` Divide by 6 `2^(3x1)=8/3 ` Take the ln of both sides and bring down the exponent `(3x1)ln2=ln(8/3) ` Simplify, `3x1=ln(8/3)/ln2 ` `3x=ln(8/3)/(ln2)+1 `...

MathSubtract 13 from both sides `8(4^(62x))=28 ` Divide by 8 `4^(62x)=7/2 ` Take the ln of both sides and bring down the exponent `(62x)ln4=ln(7/2) ` Simplify, `62x=ln(7/2)/ln4 ` `2x=ln(7/2)/ln46...

MathGiven `2^x = 3^(x+1)` applying logarithmic on both sides we get `x*ln(2) = (x+1)ln(3)` `ln(2)/ln(3) = (x+1)/x` `ln(2)/ln(3) = 1+ 1/x` `ln(2)/ln(3) 1 = 1/x` `(ln(2)  ln(3))/ln(3) = 1/x` `x =...

MathTake the ln of both sides `ln(2^(x+1))=ln(e^(1x)) ` Move the exponents out in front `(x+1)ln2=(1x)lne ` Multiply out using distributive property `xln2+ln2=lnexlne` Gather the x terms on one...

MathGiven `4^x = 5^(x^2)` applying logarithims on both sides we get `ln(4^x) = ln(5^(x^2))` => `xln(4) = x^2 (ln(5))` => `ln(4) = x (ln(5))` `x = ln(4) / ln(5) = 0.8613` or `x= 0 `

MathGiven `3^(x^2) = 7^(6  x)` applying logarithmics on both sides we get `ln(3^(x^2))=ln(7^(6  x))` => `x^2 ln(3) = (6x)ln(7)` => `x^2 ln(3) = 6*ln(7)  x* ln(7)` => `x^2 ln(3)+x*...

Math`(e^x)^24e^x5=0` Put `e^x=t.` `t^24t5=0` `t^25t+t5=0` `t(t5)+1(t5)=0` `(t5)(t+1)=0` `t=5 or t=1` `e^x = 5 and e^xgt0` Take ln on both sides, we get `x=ln5` `x=1.609`

Math`(e^x)^25e^x+6=0` Put `e^x=t` `t^25t+6=0` `t^23t2t+6=0` `t(t3)2(t3)=0` `(t2)(t3)=0` `t=2 and t=3` `e^x=2 and e^x=3.` `x=ln2 and x=ln3.` `x=0.693 and x=1.099`

Math`500=20(100e^(x/2))` `100e^(x/2)=25 ` `e^(x/2)=75 ` `e^(x/2)=75 ` Take the ln of both sides and bring down the exponent `(x/2)lne=ln75 ` Simplify, `x=8.635 `

Math`400=350(1+e^(x)) ` `1+e^(x)=8/7 ` `e^x=1/7 ` Take the ln of both sides `lne^(x)=ln(1/7) ` Bring down the exponent `xlne=ln(1/7) ` Simplify `x=1.946 `

MathTake the ln of both sides `ln(1+0.065/365)^(365t)=ln4 ` Bring down the exponent `365tln(1+0.065/365)=ln4 ` Simply `t=21.330`

MathTake the ln of both sides `ln(1.0083^(12t))=ln2 ` Bring down the exponent `12tln1.0083=ln2 ` Simplify `t=6.960 `

MathRewrite to `log_e` `log_(e)x=3` `e^3=x` `x=0.0498`

MathTake the ln of both sides `lne^x=ln2 ` Bring down the exponent `xlne=ln2 ` `x=ln2 `