Simplifying Logarithms?

Question

For some reason I'm getting really stumped on these, what should be easy, questions.

1) Ln e^7

2) e^ln3

3) 10^2log6

They need to be simplyied and I have no idea how to do them, Thanks in advance.

Answer 1

Just revise basics, right from definition.

1) lie^7 = 7lne =7...... from laws of logarithms.

2) e^ln3 = 3 .........by def.

3) 10^(2log6) = 10^log36 = 36 .....again from def.

See it the basics which are missing.

Answer 2

1) Ln e^7 = 7

2) e^ln3= 3

3) 10^2log6 = 36

Log_A X = B  is the same as A^B = X

Ln == Log_e log, base e

Ln e^7

X = e^7 A= e , so B= ? {you can see its 7}

Answer 3

These use the identities for natural logarithms

ln(e^x) = x for all x

e^ln(x) = x for all x>0

and for base 10,

10^log(x) = x for all x>0

ln e^7 = 7

using the first identity

e^ln(3) = 3

using the second identity;

and using the third:

10^(2log(6)) = 10^(log(36)) = 36

Answer 4

1 Ln (e^7) = 7

The exponential and logarithmic functions are inverses of each other, if you do one then the other, you end up with what you started.

2 e^ln3 = 3 as above

3. 10^(2log6)

now remember that alogb = log(b^a)

so we have 10^(log36) the rule for the first two questions still applies as long as the log is the same base the what is being raised so

10^(log36) =36

as ' log ' is base 10

Answer 5

1) ln e^7  = 7 ln e  = 7

2) e^ln3  =   3    

3)  10^2log6  = 10^log 6^2  = 10 ^log 36  = 36

Answer 6

Question 1

x = ln e^7

x = 7 ln e

x = 7

Question 2

x = e^(ln 3)

ln x = (ln 3)

x = 3

Question 3

Question is unclear.

Could be :-

x = 10^(2 log 6)

x = 10^(log 6²)

x = 10^log 36

log x = log 36

x = 36