Logarithms

Calculate

100

log 0.001

log 10^-3=-3

200

ln root(5)(e^2)

ln root(5)(e^2)=ln e^(2/5)=2/5

300

log_sqrt(2)32

log_sqrt(2)32=log_sqrt2 2^5=log_(2^(1/2)) (2^(1/2))^10=10

400

log_3 (root(4)(3)/sqrt(27))

log_3 (root(4)(3)/sqrt(27))=log_3 (3^(1/4)/3^(3/2))=log_3 3^(-5/4)=-5/4

500

log_(2sqrt2) 0,25

log_(2sqrt2) 0,25=log_(2^(3/2))2^-2=log_(2^(3/2))(2^(3/2))^((2/3)*(-2))=-4/3

Calculate the value of x

100

log_3 x=4

81

200

log_7 7x=2

7x=7^2->x=7^2 /7 =7

300

log_x25=-2

log_x25=-2 ->x^-2=25->x^2=1/25->x=1/5

400

log_(9/16) x=3/2

x=(9/16)^(3/2)=(sqrt(9/16))^3=(3/4)^3=27/64

500

log_(2x+3)81=2

(2x+3)^2=81->2x+3=+-9

2x+3=9->x_1=3

2x+3=-9->x_2=-6 (descartada)

Try it with another base

100

log_4 32

log_4 32=log_2 32 / log_2 4=5/2

200

log_27 (1/9)

log_27 (1/9)=log_3 (1/9) /log_3 27=-2/3

300

log_25 125 + log_4 8

log_25 125 + log_4 8=log_5 125 /log_5 25 +log_2 8 /log_2 4=3/2 +3/2=3

400

log 8 /log 2

log 8 /log 2=(log_2 8 /log_2 10) /(log_2 2 /log_2 10)=log_2 8 /log_2 2=3

500

log_(8/27)(16/81)

log_(8/27)(16/81)=log_(2/3)(16/81) /log_(2/3)(8/27)=4/3

Logarithm Rules

100

log 5 + log 20

log 5 + log 20=log (5*20)=log 100=2

200

log 2 - log 0.2

log 2 - log 0.2=log 2 - log(2/10)=log 2 - (log 2 - log 10)= 1

300

Knowing that log 2=0,3 y log 3=0,48, calculate:

log 120

log 120=log(2^2*3*10)=log 2^2+log3+log10=2*0.3+0.48+1=2.08

400

log_9 25 /log_3 5

log_9 25 /log_3 5=(log_3 25 /log_3 9)/(log_3 5/log_3 3)=(log_3 25 *log_3 3) /(log_3 9 *log_3 5)=(2log_3 5)/(2log_3 5)=1

500

log_a ac + log_d d^3 +log_b b - log_a c

log_a ac + log_d d^3 +log_b b - log_a c=log_a a+log_a c+3+1-log_a c=5

Equations

100

Log (x+3)=log 2 - log(x+2)

(x+3) =2/(x+2) ; (x+2)*(x+3)=2

x_1=4

x_2=1

200

2*log (x+5) = log (x+7)

 

log (x+5)^2 = log (x+7)->(x+5)^2=(x+7)->x^2+9x+18=0->x=-3

300

2*logx=3+log (x/10)

log x^2=log 1000 +log (x/10)->logx^2=log 100x

x^2=100x -> x=100

400

Log_4 (x+1)=log_2 (x-1)

Log_2 (x+1) /log_2 4=log_2 (x-1) ; log_2 (x+1)=log_2 (x-1)^2

(x+1)=(x-1)^2

x_1=0 (No.válida)

x_2=3

500

Log(2x-4)=-2/log(2x-4) +3

Log (2x+3)=m ; m=-2/m +3

m^2-3m+2=0

m_1=1; m_2=2

Log (2x-4)=1 ; log(2x-4)=100

x_1=7 ; x_2=52