Help me solve 6 - 2e^3x = -10.?

2 e^(3x) = 16
e^(3x) = 8
3x = ln 8
x = ln 8 / 3

6 - 2e^(3x) = -10
2e^(3x) = 6 - (-10)
e^(3x) = 16/2
3x = ln(8)
x = [ln(8)]/3

6 - 2e^3x = -10

Subtract 6 from both sides.
-2e^3x = -16

Divide both sides by -2
e^3x = 8

Use logs (ln)
ln e ^3x = 8

3x ln e = ln (8)

Divide both sides by ln e

3x = [ln ( 8 )] / ln (e)

Then divide by 3

x = { [ln (8)] / [ln (e)] } / 3

Solve

x = 0.693 (approximate)

Just like solving for any other variable, you have the same rule.

ISOLATE! (caps for emphasis, not for yelling)

Your equation can be seen as three parts, which I'll separate really big for you now...

6 - 2e^(3x) = -10

Now, when you isolate, you want to get the piece with the variable all by itself on one side of the equal sign. Notice the 6 is hanging out with it on the left hand side? So subtract that 6 over to the other side since it's currently adding to the middle term. That leaves us with...

- 2 e^(3x) = -16

Now you want to look at whether it's being multiplied by anything. In this case, it is: -2 So you'll now divide both sides by -2. That leaves us with...

e^(3x) = 8

Now you can use the definition of the logarithm or the property of logs that says if A=B then logA=logB. Meaning...

e^(3x) = 8 implies ln(e^3x) = ln(8)

There's another property of logs that says ln(e^A) = A. That means we now have...

ln(8) = ln(e^3x) = 3x implies 3x = ln(8)

Finally, by the basic Multiplicative Property of Equality, we divide both sides by 3 and obtain the following result.

x = (1/3) ln(8)

But wait, there's more. There's yet another property that states log(a^x) = x(log a). So we can go on to say

x = (1/3) ln(8) = ln(8 ^ (1/3)) = ln(2)

Subtract 6 from both sides: –2e^(3x) = –16
Divide both sides by –2: e^(3x) = 8
Take the natural logarithm of both sides: 3x = ln(8)
Divide both sides by 3: x = ln(8)/3

Your answer is: x = ln(8)/3 ≈ 0.693

6 - 2e^3x = -10
-2e^3x = -16
e^3x = 8
ln(e^3x) = ln(8)
3xln(e) = ln(8)
3x = ln(8)
x = ln(8)/3
x = .693