# Exponential equationFind the solution of the exponential equation , correct to four decimal places. e^(-4x) = 6

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We have to find the solution of : e^(-4x) = 6

e^(-4x) = 6

Take the natural logarithm of both the sides

ln[e^(-4x) = ln 6

use the property log a^b = b*log a and ln e = 1

=> -4x = ln 6

=> x = -ln 6/4

=> x = -0.4479

**The required value of x = -0.4479**

`e^(-4x) = 6 `

Take the natural log of both sides.

`ln(e^(-4x)) = ln6`

And by the property of exponents and logs, this cancels out the e.

`-4x = ln6`

Solve for x by dividing by -4 from both sides. This will get you

`x = -(ln6)/(4)`

x = .4479

We'll recall the principle that:

e^a = b <=> a = ln b

For the given equation, we'll take logarithms both sides:

ln e^(-4x) = ln 6

We'll apply the power property of logarithms:

ln e^a = a*ln e

-4x*ln e = ln 6

But ln e = 1.

-4x = ln 6

We'll divide by -4 both sides:

x = -ln 6/4

We'll get the calculator to find ln 6 = 1.7917

x = -1.7917/4

The solution of x, rounded to four decimal places, is:

x = -0.4479