Please help with equations

[gabyyyy]

Can anybody please solve step by step these two equations for x

e^(2x)=3e^x

and

3lnx=ln(3x) ?

Thanks in advance!

 

[HallsofIvy]

Can anybody please solve step by step these two equations for x

e^(2x)=3e^x

A first step could be to divide both sides by e^x:
e^x= 3. Now take the natural logarithm of both sides.

and

3lnx=ln(3x) ?

Thanks in advance!

You need to know some basic properties of logarithms! log(3x)= log(x)+ log(3) so your equation can be written
3ln(x)= ln(x)+ ln(3)
Subtract ln(3) from both sides:
2ln(x)= ln(3)
Divide both sides by 2:
ln(x)= ln(3)/2

Use another property of logarithms: log(3)/2 = log(3^(1/2))= log(sqrt(3))
so ln(x)= ln(sqrt(3)).

Finally use the fact that logarithm is "one to one":
x= sqrt(3).

 

[gabyyyy]

An obvious first step should be to divide both sides by e^x:
e^x= 3. Now take the natural logarithm of both sides.


You need to know some basic properties of logarithms! log(3x)= log(x)+ log(3) so your equation can be written
3ln(x)= ln(x)+ ln(3)
Subtract ln(3) from both sides:
2ln(x)= ln(3)
Divide both sides by 2:
ln(x)= ln(3)/2

Use another property of logarithms: log(3)/2 = log(3^(1/2))= log(sqrt(3))
so ln(x)= ln(sqrt(3)).

Finally use the fact that logarithm is "one to one":
x= sqrt(3).

Thank you so much! I really appreciate it!

 

[david_the_tutor]

Natural log equations

Can anybody please solve step by step these two equations for x

e^(2x)=3e^x

and

3lnx=ln(3x) ?

Thanks in advance!

e^2x = 3e^x
e^x(e^x) = e^x(3) --------- Factoring out GCF, e^x
Thus, e^x = 3
ln 3 = x ------ Converting to LOGARITHMIC form
x = ln(3) = 1.098612


3ln x = ln (3x)
ln x³ = ln 3x
Therefore, x³ = 3x
x(x²) = x(3) ------- Factoring out GCF, x
x² = 3
x = ± sqrt(3)
x = 1.732051, or - 1.732051 (ignore NEGATIVE VALUE as this is an EXTRANEOUS solution)