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MathBring up the exponent, `log_(6)(2x)^(4) `

MathBring up the exponent, `logxlog(x+1)^2 ` Condense by division `log(x/(x+1)^2) `

MathBring up the exponents in both terms `ln8^2+ln(z4)^5 ` ` ` Condense by multiplication `ln(8^2)(z4)^2 ` Simplify, `ln64(z4)^2 `

MathBring up the exponents `logxlogy^2+logz^3 ` ` ` `log(x/y^2)+logz^3 ` `log((xz^3)/y^2) `

MathBring up the exponents `log_3x^3+log_3y^4log_3z^4` ` ` Condense using multiplication and division `log_3(x^3*y^4)log_3z^4 ` `log_(3)((x^3y^4)/(z^4)) `

MathCondense using multiplication and division `lnxln((x+1)/(x1)) ` `ln(x/((x+1)/(x1))) ` Simplify, `ln((x(x1))/(x+1)) `

Math`4lnz(z+5)ln(z5)^2` `ln(z(z+5))^4ln(z5)^2 ` `ln(((z(z+5)^4))/(z5)^2) `

Math`(1/3)(ln(x+3)^2+lnxln(x^21)) ` `(1/3)(ln(((x+3)^2)x)/(x^21)) ` `ln(((x+3)^2x)/(x^21))^(1/3) `

Math`2(lnx^3ln(x+1)ln(x1))` `2(lnx^3ln((x+1)/(x1)))` `2(ln(x^3)/((x+1)/(x1))) ` `ln((x^(3)*(x1))/(x+1))^2 `

MathThe formula to change the base of the log would be so that would translate to : = `logx/log2` ` `Refer to the attachement below for the graph.Hope this helped!

MathExpand using addition (log multiplication) and subtraction (log division) `lnx^2+ln(sqrt(y/z)) ` ` ` `2lnx+ln((sqrty)/(sqrtz)) ` `2lnx+(1/2)lny(1/2)lnz `

MathExpand using addition (log multiplication) and subtraction (log division) `log_2x^4+log_(2)((sqrty)/(sqrt(z^3))) ` Simplify and bring down the exponents `4log_2x+(1/2)log_2y(3/2)log_2z `

MathExpand using addition (log multiplication) and subtraction (log division) `log_5x^2(log_5y^2+log_5z^3) ` `2log_5x(2log_5y+3log_5z) `

MathExpand using addition (log multiplication) and subtraction (log division) `(log_10x+log_10y^4)log_10z^5 ` `(log_10x+4log_10y)5log_10z `

Math`log_b10=log_b(2*5) ` `log_b2+log_b5=log_b10 ` `0.3562+0.8271=log_b10 ` `log_b10=1.1833 `

Math`log_b(2/3)=log_b2log_b3 ` `log_b(2/3)=0.35620.5646 ` `log_b(2/3)=0.2084 `

Math`log_b8=log_b2^3 ` `log_b8=3log_b2 ` `log_b8=3(0.3562) ` `log_b8=1.0686 `

Math`log_bsqrt2=log_b2^(1/2) ` `log_bsqrt2=(1/2)log_b2 ` `log_bsqrt2=(1/2)0.3562 ` `log_bsqrt2=0.1781 `

Math`log_b45=log_b(3*3*5) ` `log_b45=log_b3+log_b3+log_b5 ` `log_b45=0.5646+0.5646+0.8271 ` `log_b45=1.9563 `

MathCondense using multiplication` ` Given, `ln2+lnx ` Then, condensed it is, `ln2x ` ` `

MathCondense using division, Given, `log_5(8)log_5t ` Therefore, it becomes `log_5(8/t) `

MathFirst, bring up the exponents, `log_2x^2+log_2y^2 ` ` ` Condense using multiplication `log_2x^2y^2 `

MathAll you need to do is bring up the exponent `log_7(z2)^(2/3) `

MathBring down the exponents, `2lne+5lne ` Remember that `lne=1 ` Therefore, `2(1)+5(1) ` Simplify, `7 `

MathBring down the exponents, `12lne5lne ` Remember that Therefore, `12(1)5(1) ` Simplify, `7 `

MathGiven, `log_5(75)log_5(3) ` Condense `log_5(25) ` Set equal to x `log_5(25)=x ` `5^x=25 ` `5^x=5^2 ` `x=2 `

MathGiven, `log_4(2)+log_4(32) ` Condense, `log_4(64) ` Set equal to x `log_4(64)=x ` `4^x=64 ` `4^x=4^3 ` `x=3 `

MathExpand using addition, `ln4x=ln4+lnx `

MathExpand using addition, `log_3(10z)=log_3(10)+log_3z `

MathBring down the exponent, `4log_8x ` Done. You cannot expand this log any further.

MathExpand by using subtraction, `log_10(y/2)=log_10ylog_10(2) `

MathExpand using subtraction, `log_5(5/x)=log_5(5)log_5x `

MathExpand using subtraction, `log_6(1/z^3)=log_6(1)log_6z^3 ` Bring down the exponent, `log_6(1/z^3)=log_6(1)3log_6z = 0  3log_6z`

MathBring down the exponent, `lnsqrtz=lnz^(1/2) ` Therefore, `lnsqrtz=(1/2)lnz `

MathRewrite as `ln(t^(1/3))` Bring down the exponent, `(1/3)lnt`

MathSince all of the terms are being multiplied inside the ln, that means to expand them, one must rewrite them as added. Therefore, Given `lnxyz^2 ` Then, `lnxyz^2=lnx+lny+2lnz `

MathSince all of the terms are multiplied, one can just expand by adding them all. Given, `log(4x^2y) ` Then, Separate all of the terms and bring the exponents down in front where necessary....

MathSince all of the terms are multiplied together, one can expand by adding them together. Given, `ln((z)(z1)^2)` Then, ` ln((z)(z1)^2)=lnz+2ln(z1) `

MathExpand using subtraction because the terms are being divided in the ln. Given, `ln((x^21)/(x^3)) ` Then, after expanded and bringing down the exponents where necesarry `ln((x^21)/(x^3))=...

MathExpand using subtraction because the terms inside the log are being divided. Given, `log_2(sqrt(a1)/9) ` Then, `log_2(sqrt(a1)/9)=log_2(a1)^(1/2)log_2(9) ` Therefore, after bring down the...

MathExpand by using subtraction, Given, `ln(6/sqrt(x^2+1)) ` Then, expanding and bringing down the exponent where necessary, `ln(6/sqrt(x^2+1))=ln6(1/2)ln(x^2+1) `

MathExpand using subtraction because the terms are being divided inside the ln. Given, `ln(root(3)(x/y)) ` Then, `ln(root(3)(x/y))=ln(root(3)x)ln(root(3)y)` Therefore, after bringing the exponent...

MathExpand by subtracting the terms because they are divided inside of the ln. Given, `ln(x^2/y^3) ` Then, ` ln(x^2/y^3) =lnx^2lny^3` Bring down the exponents, `ln(x^2/y^3) =2lnx3lny`

MathYou need to use a change of base formula to solve this. Use `2` since both `4` and `8` and powers of `2` `` `log_4(8)=((log_2(8))/(log_2(4))`) This simplifies down to, `3/2`

Math`log_5 (1/250) = log_5(1/125) + log_5(1/2)` `log_5 (1/250) = log_5 (5^3) + log_5 (2^1) ` `log_5 (1/250) = 3 + 1*log_5(2) ` `log_5 (1/250) = 3  log_5(2) `

Math`log_10(9/300) = log_10(3/100) ` `log_10(3/100)= log_10(3)  log_10(100) ` `log_10(3/100)= log_10(3)  log_10(10^2) ` `log_(3/100)= log_(3)  2 `

MathExpand using addition`ln5 + ln(e^6) ` bring down the exponent`ln5 + 6ln(e) ` Remember that `lne=1 ` `ln5 + 6 `

Math`ln(6/e^2) ` When you divide inside the ln, you subtract when you expand`ln(6)  ln(e^2) ` Bring down the exponent and remember than `lne=1 ` `ln(6)  2 `

MathSet `log_3(9)=x` Rewrite, `3^x=9 ` `x=2 `

Math`log_5(1/125)=x ` Rewrite log5 1/125 : `1/125=125^1 ` This gives: `log_5(125^1) ` , bring down the power `1(log_5(125)) ` , rewrite 125 in term of power of 5 `1(log_5(5^3)) ` , again...