Logs and natural logs question ...please ? thanks

Question

Let u=log 2 and v=log3. evaluate the following expressions in terms of u and/or v.

1) log 0.08

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2) log₃ 3^5x+1=2-----solve for x

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Simplify the expression if possible

3) e^{-3ln t}

4) 10^{2+log x}

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Solve for x

5) log(2x+7)=2

6) log(2x)=log(x+10)

Answer 1

1) log(0.08)=log(2*0.04)

log(2*0.04)=log(2)+log(0.04)

u+log(0.04)

u+(-1.39794), or

log(2)=0.30102999

0.30102999-1.39794000=-1.09691001, or

log(8*10^-2)

log(8)+log(10^-2)=log(2^3)+log(10^-2)

3log(2)+(-2)

if u=log(2),then 3u-2

 

2) log3 3^5x+1=2

(5x+1)log3 3=2

log3 3=2/(5x+1)

3^(2/5x+1)=3

2/(5x+1)=1

2=5x+1

1=5x

x=1/5

 

5) log(2x+7)=2

  10^2=2x+7

     100=2x+7

       93=2x

         x=93/2

 

6) log(2x)=log(x+10)

          2x=x+10

            x=10

Answer 2

1) log 0.08 = log ( 8 X 10 ^-2) = log 8 + log 10^-2 =

 

                                          = log 2³ + -2

                                           = 3 log 2 -2 = 3U -2

 

  log₃ 3^ ( 5X +1 )  = 2

 

  ( 5X +1 ) log ₃3 =  2

 

     5X +1 =2       X = 1/5

 

3 )

    e ^-3lnt

 

 ln ( e ^-3lnt ) = - 3ln t

 

                    = ln 1/t^3

 

   Then:

               e ^-3lnt= 1/ t^3

 

   4 )

 

       10 ^{2 + log X}

 

   log ( 10^{2 + log X} ) = 2 + log X

 

     10 ^{2 + log X} = 2 + log X

 

   5) log ( 2X + 7 ) = 2

 

        2X + 7 = 100

 

         X = 93/2

 

   6 ) log 2X = log ( X +10)

 

            2X = X +10

 

           X = 10