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

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Given the equation e^(-4x) = 6.

We need to find x values where the equality holds.

To solve the exponent equation, will apply the natural logarithm to both sides:

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

Now we know from logarithm properties that:

ln a^b = b*ln a

==> (-4x) * ln e = ln 6

But ln e = 1

==> -4x *1 = ln 6

==> -4x = ln 6

Now divide by -4:

==> x = ln 6 / -4

**==> x = -0.4479**

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**

To find the solution to e^(-4x) = 6.

Solution:

e^(-4x) = 6.

We take natural logarithms of both sides:

-4x = ln 6.

-4x = 1.791759 469.

We divide both sides by -4:

x = 1.791759 469/-4

x= -0.447939...

x = -0.4479 for 4 decimal places.