Proving y= 10^x Using Logarithms to Base e

[lionely]

Homework Statement

If y= 10^x, show by taking logarithms to base e that y = e^xln10

Homework Equations

The Attempt at a Solution

Well what I did was y= 10^x so ln y = xln10

they told me that y = e^xln10
so ln y = xln 10 , so ln y = lny

so y = y
so y= 10^x.

The way I did it doesn't feel right to me . Is there another approach?

 

[SteamKing]

Homework Statement

If y= 10^x, show by taking logarithms to base e that y = e^xln10

Homework Equations

The Attempt at a Solution

Well what I did was y= 10^x so ln y = xln10

they told me that y = e^xln10
so ln y = xln 10 , so ln y = lny

so y = y
so y= 10^x.

The way I did it doesn't feel right to me . Is there another approach?

You seem to have gotten in a loop with this step:

so ln y = xln 10 , so ln y = lny

so y = y
so y= 10^x

This is the last meaningful equation you have:

ln y = x ln 10

How do you go about getting rid of the logarithms on both sides? That will lead you to your answer.

 

[Mark44]

Homework Statement

If y= 10^x, show by taking logarithms to base e that y = e^xln10

Homework Equations

The Attempt at a Solution

Well what I did was y= 10^x so ln y = xln10

Now make each side of the equation the exponent on e.

they told me that y = e^xln10
so ln y = xln 10 , so ln y = lny

so y = y
so y= 10^x.

The way I did it doesn't feel right to me . Is there another approach?

 

[lionely]

you mean do e^y = e^(10^x) ?

 

[Tanya Sharma]

log_ca = b is same as writing c^b = a . For example log₁₀1000 = 3 is equivalent to 10³ = 1000 .

Now use this fact .You are just one step away.

 

[Mark44]

you mean do e^y = e^(10^x) ?

No.
You had ln(y) = xln(10).
Now write each side as the exponent on e. The idea is that if A = B, then e^A = e^B.

 

[lionely]

Thank you guys!