Solve the exponential equation. Express the solution set in terms of natural logarithms.
4^(x + 4) = 5^(2x + 5)-This question I don't understand.
Alright. I think I might have gotten the answer. Write back and tell me if you think it's right. If you think I messed up, I'll keep trying.
First use ln to rearrange the original:
4^(x+4)=5^(2x+5)
ln4^(x+4)=ln5^(2x+5)
Put the exponents in front because of the properties of logs.
(x+4)(ln4)=(2x+5)(ln5)
Rearrange to get all the x's on one side.
(x+4)/(2x+5)=(ln5)/(ln4)
I'm sure you got to this point. But after here, it starts to get a little tricky.
Technically, (ln5)/(ln4) is equal to [(ln5)/(ln4)]/1 right?
Let's set a variable......how about "u" equal to the numerator of this fraction.
So... u=[(ln5)/(ln4)]
Let's plug that back in to what we had from before:
(x+4)/(2x+5)=u/1
Use cross multlipication to get this:
(x+4)=(2x+5)(u)
Distribute the u
x+4=2xu+5u
Get all the parts with x's on one side:
x-2xu=5u-4
Pull out an x:
x(1-2u)=5u-4
Divide the (5u-4) by (1-2u) to get x by itself:
x=(5u-4)/(1-2u)
Plug the u value back in and simplify
x=(5((ln5)/(ln4))-4) / (1-2((ln5)/(ln4))
I think that's as far as you can go.
Good luck! :-)
the closest answers that I have that are given in the multiple choice is {5 ln 5 - 4 ln 4/ln 4-2 ln 5}
Take the natural logs (ln) of both sides.
(x+4) ln 4 = (2x+5) ln 5
x ln 4 + 4 ln 4 = 2x ln 5 + 5 ln 5
x (2 ln 5 - ln 4) = 4 ln 4 - 5 ln 5
x = (4ln4 - 5ln5)/(2ln5 - ln4)
This becomes the same as your answer if you multiply both numerator and denominiator by -1
To solve the exponential equation 4^(x + 4) = 5^(2x + 5), we can take the natural logarithm (ln) of both sides of the equation.
ln(4^(x + 4)) = ln(5^(2x + 5))
Using the logarithmic power rule, we can bring down the exponent:
(x + 4)ln(4) = (2x + 5)ln(5)
Next, simplify by distributing ln(4) and ln(5) to their respective terms:
xln(4) + 4ln(4) = 2xln(5) + 5ln(5)
Now, rearrange the equation to isolate the terms with x:
xln(4) - 2xln(5) = 5ln(5) - 4ln(4)
Factor out x:
x(ln(4) - 2ln(5)) = 5ln(5) - 4ln(4)
Finally, divide both sides by (ln(4) - 2ln(5)):
x = (5ln(5) - 4ln(4)) / (ln(4) - 2ln(5))
So, the solution to the exponential equation 4^(x + 4) = 5^(2x + 5) is x = (5ln(5) - 4ln(4)) / (ln(4) - 2ln(5)).