Ok, I was getting all my answers right until I stumbled on this one:
5e^0.2x=7
This is what I was doing:
5e^0.2x=7
5lne^0.2x=ln7
5*0.2x=ln7
0.2x=(ln7/5)
???
The answer is 5ln(7/5) but i do not know how they derived to this. Thank you so much!
What you have so far is correct.
you would end up with
x = ln(7/5)/.2 which is the same as their answer.
You just have a different form.
notice you are dividing by .2 or 1/5
and isn't division by 1/5 the same as multiplying by 5?
there you go!
ln(7/5)/.2 = 5ln(7/5)
To solve the equation 5e^(0.2x) = 7, you are on the right track. Let me guide you on how to proceed further:
Step 1: Divide both sides of the equation by 5 to isolate the exponential term:
e^(0.2x) = 7/5
Step 2: Take the natural logarithm (ln) of both sides of the equation to eliminate the exponential term:
ln(e^(0.2x)) = ln(7/5)
Step 3: Apply the property of logarithms, which states that ln(e^a) = a. This simplifies the left side of the equation:
0.2x = ln(7/5)
Step 4: Divide both sides of the equation by 0.2 to solve for x:
x = ln(7/5) / 0.2
Now, to simplify the expression ln(7/5) / 0.2, we can make use of the logarithmic property that ln(a/b) = ln(a) - ln(b):
x = ln(7) - ln(5) / 0.2
Finally, simplify further by multiplying the numerator and denominator by 5 to get rid of the fraction:
x = (5ln(7) - 5ln(5)) / 0.2
This can be further simplified by factoring out the common factor of 5:
x = 5(ln(7) - ln(5)) / 0.2
Now, simplifying the expression inside the parentheses:
x = 5ln(7/5) / 0.2
And this is equivalent to:
x = 5ln(7/5)
So, the solution to the equation 5e^(0.2x) = 7 is x = 5ln(7/5).