compress the expression into a single logarithm with no coefficent:

Question

show step by step work:

3ln2+2ln5-ln10

Answer 1

To compress any logarithmic functions (log, ln), here are some helpful notes:

Any coefficient in front of a logarithm is considered to be equivalent to an exponent of the argument.

(That's fancy talk to mean the number in front of the function is the exponent of the number inside.)

EX: 5log(3)=log(3^5)

NOTE: This step should be done before the other two! (Similar to PEMDAS)

Anytime logarithms are added, this is considered to be the equivalent of their arguments being multiplied.

EX: ln(3)+ln(5)=ln(3*5)=ln(15)

Anytime logarithms are subtracted, this is considered to be the equivalent of their arguments being divided.

EX: log(6)-log(2)=log(6/2)=log(3)

Now for the problem at hand: 3ln2+2ln5-ln10

Bring the coefficients in as exponents:

3ln2+2ln5-ln10 = ln(2^3)+ln(5^2)-ln10

Simplify:

ln(2^3)+ln(5^2)-ln10 = ln(8)+ln(25)-ln10

Add and subtract the logarithms(Multiply and/or Divide arguments):

ln(8)+ln(25)-ln10 = ln(8*25/10)

Simplify:

ln(8*25/10) = ln(20)

ANS: ln(20)