Could someone help me with these please ? I'm a little confused

Question

Express the following in terms of x without logs.

1)ln e^2x

2)e^ln(3x+2)

3)ln(1/e^5x)

4)ln sqrt(e^x)

Answer 1

Hi!

 

the trick is to realize that ln is log base e

 

1. Ln(e^2x) = loge(e^2x)

 

the easiest way I understand it is through examples. Let's use 10 instead of e

 

log10(10^2x)

if it were 10(10^2) then the answer is 2!

 

therefore, loge(e^2x) = 2x

 

 

i will use this same method going forward:

 

2. E^loge(3x+2)

10^log10(3x+2)

10'to the what = 3x+2?

well it's not important because 10 is then raised to that "what", equaling 3x+2

therefore the answer is

3x+2

 

3. Loge(1/e^5x) = loge(e^-5x)

log10(10^-5x) = -5x

therefore

-5x

 

4. Loge(sqrt(e^x)) = loge((e^x)^1/2)

log10(sqrt(10^x)) = log10((10^x)^1/2)

Ex. 3^2^2 = 3^4

therefore

10^x^1/2 = 10^x/2

log10(10^x/2) = x/2

therefore

x/2

 

 

 

 

 

 

 

 

 

Answer 2

1)ln e^2x

 

The natural logarithm of the exponential function equals the exponent->ln(e^b)=b

therefore,

ln e^2x=2x

 

2)e^ln(3x+2)

The exponential function of the natural logarithm equals the exponent->e^ln(b)=b

therefore,

e^ln(3x+2)=3x+2

 

3)ln(1/e^5x)

A negative exponent equals the reciprocal of the number->e^-b=1/(e^b)

therefore,

ln(1/e^5x)=ln((e^5x)^-1))

 

(e^b)^c = e^(b*c)

therefore,

ln((e^5x)^-1))=ln(e^-5x)

 

from rule 1,

ln(e^-5x)=-5x

 

 

4)ln sqrt(e^x)

A sqrt of a number equals the number raised to 1/2 power -> sqrt(e^b)=e^b/2

therefore,

ln sqrt(e^x)=ln (e^x/2)

From rule 1, 

ln (e^x/2)=x/2