logb(23)logb(23) Approximate the logarithm using the properties of logarithms

# logb(23)logb(23) Approximate the logarithm using the properties of logarithms

• MATH

y=3cot(πx2)y=3cot(πx2) Sketch the graph of the function. (Include two full periods.)

Graph

• MATH

y=3cot(2x)y=3cot(2x) Sketch the graph of the function. (Include two full periods.)

Graph

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y=csc(x3)y=csc(x3) Sketch the graph of the function. (Include two full periods.)

Graph

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y=csc(x2)y=csc(x2) Sketch the graph of the function. (Include two full periods.)

Graph

• MATH

y=−2sec(4x)+2y=-2sec(4x)+2 Sketch the graph of the function. (Include two full periods.)

Graph

• MATH

y=(12)sec(πx)y=(12)sec(πx) Sketch the graph of the function. (Include two full periods.)

Make a graph of the function (refer to the attached image)

• MATH

y=3csc(4x)y=3csc(4x) Sketch the graph of the function. (Include two full periods.)

Graph

• MATH

y=csc(πx)y=csc(πx) Sketch the graph of the function. (Include two full periods.)

Make a graph of the function (refer to the attached image)

• MATH

y=(14)sec(x)y=(14)sec(x) Sketch the graph of the function. (Include two full periods.)

Graph

• MATH

y=(−12)sec(x)y=(-12)sec(x) Sketch the graph of the function. (Include two full periods.)

Graph

• MATH

y=−3tan(πx)y=-3tan(πx) Sketch the graph of the function. (Include two full periods.)

Make a graph of the function (see the attached image) To find the period, apply the formula for tangent and cotangent, which is , where b is what is attached to the x. Plugging in what you know,…

• MATH

y=−2tan(3x)y=-2tan(3x) Sketch the graph of the function. (Include two full periods.)

Graph

• MATH

y=tan(4x)y=tan(4x) Sketch the graph of the function. (Include two full periods.)

Graph

• MATH

y=(13)tan(x)y=(13)tan(x) Sketch the graph of the function. (Include two full periods.)

Graph

• MATH

f(x)=log11.8(x)f(x)=log11.8(x) Use the change-of-base formula to rewrite the logarithm as a ratio of…

We have the change-of-base formula as: logb(x)=log(x)log(b)logb(x)=log(x)log(b) Similarly, we can write, log11.8(x)=log(x)log(11.8)log11.8(x)=log(x)log(11.8) = log(x)1.07log(x)1.07 The graph is shown below.

• MATH

f(x)=log14(x)f(x)=log14(x) Use the change-of-base formula to rewrite the logarithm as a ratio of…

We have the change-of-base formula as: logb(x)=log(x)log(b)logb(x)=log(x)log(b) Using this, we can write, log14(x)=log(x)log(14)log14(x)=log(x)log(14) =log(x)−6.021log(x)-6.021 The graph is shown…

• MATH

f(x)=log12(x)f(x)=log12(x) Use the change-of-base formula to rewrite the logarithm as a ratio of…

We have the change-of-base formula as: logb(x)=log(x)log(b)logb(x)=log(x)log(b) Similarly, we can write, log12(x)=log(x)log12=log(x)−0.301log12(x)=log(x)log12=log(x)-0.301 The graph is shown below.

• MATH

f(x)=log2(x)f(x)=log2(x) Use the change-of-base formula to rewrite the logarithm as a ratio of…

The formula to change the base of the log would be so that would translate to : = logxlog2logxlog2 Refer to the attachement below for the graph.Hope this helped!

• MATH

(12)(log4(x+1)+2log4(x−1))+6log4(x)(12)(log4(x+1)+2log4(x-1))+6log4(x) Condense the expression to the…

Let us first see some of the properties of a logarithm. log(x)+log(y)=log(xy) alog(x)=log(xaxa ) Using these, we can solve the equation. We have the equation as:…

• MATH

(13)(log8(y)+2log8(y+4))−log8(y−1)(13)(log8(y)+2log8(y+4))-log8(y-1) Condense the expression to the logarithm…

Let us first see some of the properties of a logarithm. log(x)+log(y)=log(xy) log(x)-log(y)=log(xy)log(xy) alog(x)=log(xa)xa) Using these, we can solve the above equation as:…

• MATH

2(3ln(x)−ln(x+1)−ln(x−1))2(3ln(x)-ln(x+1)-ln(x-1)) Condense the expression to the logarithm of a single…

2(lnx3−ln(x+1)−ln(x−1))2(lnx3-ln(x+1)-ln(x-1)) 2(lnx3−ln(x+1x−1))2(lnx3-ln(x+1x-1)) 2(ln(x3)x+1x−1)2(ln(x3)x+1x-1) ln(x3⋅(x−1)x+1)2ln(x3⋅(x-1)x+1)2

• MATH

(13)(2ln(x+3)+ln(x)−ln(x2−1))(13)(2ln(x+3)+ln(x)-ln(x2-1)) Condense the expression to the logarithm of a…

(13)(ln(x+3)2+lnx−ln(x2−1))(13)(ln(x+3)2+lnx-ln(x2-1)) (13)⎛⎝⎜ln(((x+3)2)x)x2−1⎞⎠⎟(13)(ln(((x+3)2)x)x2-1) ln((x+3)2xx2−1)13ln((x+3)2xx2-1)13

• MATH

4(ln(z)+ln(z+5))−2ln(z−5)4(ln(z)+ln(z+5))-2ln(z-5) Condense the expression to the logarithm of a single…

4lnz(z+5)−ln(z−5)24lnz(z+5)-ln(z-5)2 ln(z(z+5))4−ln(z−5)2ln(z(z+5))4-ln(z-5)2 ln⎛⎝⎜(z(z+5)4)(z−5)2⎞⎠⎟ln((z(z+5)4)(z-5)2)

• MATH

ln(x)−(ln(x+1)+ln(x−1))ln(x)-(ln(x+1)+ln(x-1)) Condense the expression to the logarithm of a single quantity.

Condense using multiplication and division lnx−ln(x+1x−1)lnx-ln(x+1x-1) ln(xx+1x−1)ln(xx+1x-1) Simplify, ln(x(x−1)x+1)ln(x(x-1)x+1)

• MATH

3log3(x)+4log3(y)−4log3(z)3log3(x)+4log3(y)-4log3(z) Condense the expression to the logarithm of a single…

Bring up the exponents log3x3+log3y4−log3z4log3x3+log3y4-log3z4  Condense using multiplication and division log3(x3⋅y4)−log3z4log3(x3⋅y4)-log3z4 log3(x3y4z4)log3(x3y4z4)

• MATH

log(x)−2log(y)+3log(z)log(x)-2log(y)+3log(z) Condense the expression to the logarithm of a single quantity.

Bring up the exponents logx−logy2+logz3logx-logy2+logz3  log(xy2)+logz3log(xy2)+logz3log(xz3y2)log(xz3y2)

• MATH

2ln(8)+5ln(z−4)2ln(8)+5ln(z-4) Condense the expression to the logarithm of a single quantity.

Bring up the exponents in both terms ln82+ln(z−4)5ln82+ln(z-4)5  Condense by multiplication ln(82)(z−4)2ln(82)(z-4)2 Simplify, ln64(z−4)2ln64(z-4)2

• MATH

log(x)−2log(x+1)log(x)-2log(x+1) Condense the expression to the logarithm of a single quantity.

Bring up the exponent, logx−log(x+1)2logx-log(x+1)2 Condense by division log(x(x+1)2)log(x(x+1)2)

• MATH

−4log6(2x)-4log6(2x) Condense the expression to the logarithm of a single quantity.

Bring up the exponent, log6(2x)−4log6(2x)-4

• MATH

(14)log3(5x)(14)log3(5x) Condense the expression to the logarithm of a single quantity.

Bring up the exponent, log3(5x)14log3(5x)14

• MATH

(23)log7(z−2)(23)log7(z-2) Condense the expression to the logarithm of a single quantity.

All you need to do is bring up the exponent log7(z−2)23log7(z-2)23

• MATH

2log2(x)+4log2(y)2log2(x)+4log2(y) Condense the expression to the logarithm of a single quantity.

First, bring up the exponents, log2x2+log2y2log2x2+log2y2  Condense using multiplication log2x2y2log2x2y2

• MATH

log5(8)−log5(t)log5(8)-log5(t) Condense the expression to the logarithm of a single quantity.

Condense using division, Given, log5(8)−log5tlog5(8)-log5t Therefore, it becomes log5(8t)log5(8t)

• MATH

ln(2)+ln(x)ln(2)+ln(x) Condense the expression to the logarithm of a single quantity.

Condense using multiplication Given, ln2+lnxln2+lnx Then, condensed it is, ln2xln2x

• MATH

logb(3b−−√3)logb(3b3) Approximate the logarithm using the properties of logarithms, given…

logb((3b)13)=(13)logb(3b)logb((3b)13)=(13)logb(3b) =(13)logb(3)+(13)logb(b)=(13)logb(3)+(13)logb(b) =(13)(0.5646+1)=(13)(0.5646+1) =(1.5646/3) =0.5215

• MATH

logb(3b2)logb(3b2) Approximate the logarithm using the properties of logarithms, given

logb(3b2)=logb(3)+logb(b2)logb(3b2)=logb(3)+logb(b2) =logb(3)+2logb(b)=logb(3)+2logb(b) =logb(3)+2=logb(3)+2 =0.5646+2 =2.5646

• MATH

logb((2b)−2)logb((2b)-2) Approximate the logarithm using the properties of logarithms, given

logb(2b)−2=(−2)logb(2b)logb(2b)-2=(-2)logb(2b) =(−2)(logb(2)+logb(b))=(-2)(logb(2)+logb(b)) =(−2)(0.3562+1)=(-2)(0.3562+1) =−2.7124=-2.7124

• MATH

logb(45)logb(45) Approximate the logarithm using the properties of logarithms, given

logb45=logb(3⋅3⋅5)logb45=logb(3⋅3⋅5) logb45=logb3+logb3+logb5logb45=logb3+logb3+logb5 logb45=0.5646+0.5646+0.8271logb45=0.5646+0.5646+0.8271 logb45=1.9563logb45=1.9563

• MATH

logb(2–√)logb(2) Approximate the logarithm using the properties of logarithms, given

logb2–√=logb212logb2=logb212 logb2–√=(12)logb2logb2=(12)logb2 logb2–√=(12)0.3562logb2=(12)0.3562logb2–√=0.1781logb2=0.1781

• MATH

logb(8)logb(8) Approximate the logarithm using the properties of logarithms, given

logb8=logb23logb8=logb23 logb8=3logb2logb8=3logb2 logb8=3(0.3562)logb8=3(0.3562) logb8=1.0686logb8=1.0686

• MATH

logb(23)logb(23) Approximate the logarithm using the properties of logarithms, given

logb(23)=logb2−logb3logb(23)=logb2-logb3 logb(23)=0.3562−0.5646logb(23)=0.3562-0.5646 logb(23)=−0.2084logb(23)=-0.2084

• MATH

logb(10)logb(10) Approximate the logarithm using the properties of logarithms, given

logb10=logb(2⋅5)logb10=logb(2⋅5) logb2+logb5=logb10logb2+logb5=logb10 0.3562+0.8271=logb100.3562+0.8271=logb10 logb10=1.1833logb10=1.1833

• MATH

ln((x2)(x+2)−−−−−−−−−√)ln((x2)(x+2)) Use the properties of logarithms to expand the expression as a sum,…

ln(x2(x+2))12=(12)ln(x2(x+2))ln(x2(x+2))12=(12)ln(x2(x+2)) =(12)lnx2+(12)ln(x+2)=(12)lnx2+(12)ln(x+2) =(22)lnx+(12)ln(x+2)=(22)lnx+(12)ln(x+2) =lnx+(12)ln(x+2)=lnx+(12)ln(x+2)

• MATH

ln((x3)(x2+3)−−−−−−−−−−√4)ln((x3)(x2+3)4) Use the properties of logarithms to expand the expression as a…

ln((x3)(x2+3))14=(14)ln(x3(x2+3))ln((x3)(x2+3))14=(14)ln(x3(x2+3)) =(14)lnx3+(14)ln(x2+3)=(14)lnx3+(14)ln(x2+3) =(34)lnx+(14)ln(x2+3)=(34)lnx+(14)ln(x2+3)

• MATH

log10(xy4z5)log10(xy4z5) Use the properties of logarithms to expand the expression as a sum,…

Expand using addition (log multiplication) and subtraction (log division) (log10x+log10y4)−log10z5(log10x+log10y4)-log10z5 (log10x+4log10y)−5log10z(log10x+4log10y)-5log10z

• MATH

log5(x2(y2)(z3))log5(x2(y2)(z3)) Use the properties of logarithms to expand the expression as a…

Expand using addition (log multiplication) and subtraction (log division) log5x2−(log5y2+log5z3)log5x2-(log5y2+log5z3) 2log5x−(2log5y+3log5z)2log5x-(2log5y+3log5z)

• MATH

log2((x4)yz3−−−√)log2((x4)yz3) Use the properties of logarithms to expand the expression as a…

Expand using addition (log multiplication) and subtraction (log division) log2x4+log2(y√z3−−√)log2x4+log2(yz3) Simplify and bring down the exponents 4log2x+(12)log2y−(32)log2z4log2x+(12)log2y-(32)log2z

• MATH

ln((x2)yz−−√)ln((x2)yz) Use the properties of logarithms to expand the expression as a sum,…

Expand using addition (log multiplication) and subtraction (log division) lnx2+ln(yz−−√)lnx2+ln(yz)  2lnx+ln(y√z√)2lnx+ln(yz) 2lnx+(12)lny−(12)lnz2lnx+(12)lny-(12)lnz

• MATH

ln(x2y3)ln(x2y3) Use the properties of logarithms to expand the expression as a sum,…

Expand by subtracting the terms because they are divided inside of the ln. Given, ln(x2y3)ln(x2y3) Then, ln(x2y3)=lnx2−lny3ln(x2y3)=lnx2-lny3 Bring down the exponents, ln(x2y3)=2lnx−3lnyln(x2y3)=2lnx-3lny

• MATH

ln(xy−−√3)ln(xy3) Use the properties of logarithms to expand the expression as a sum,…

Expand using subtraction because the terms are being divided inside the ln. Given, ln(xy−−√3)ln(xy3) Then, ln(xy−−√3)=ln(x−−√3)−ln(y√3)ln(xy3)=ln(x3)-ln(y3)Therefore, after bringing the exponent…