We have been given an equation [tex]ae^{ct}=d[/tex]. We are asked to solve the equation for t.
First of all, we will divide both sides of equation by a.
[tex]\frac{ae^{ct}}{a}=\frac{d}{a}[/tex]
[tex]e^{ct}=\frac{d}{a}[/tex]
Now we will take natural log on both sides.
[tex]\text{ln}(e^{ct})=\text{ln}(\frac{d}{a})[/tex]
Using natural log property [tex]\text{ln}(a^b)=b\cdot \text{ln}(a)[/tex], we will get:
[tex]ct\cdot \text{ln}(e)=\text{ln}(\frac{d}{a})[/tex]
We know that [tex]\text{ln}(e)=1[/tex], so we will get:
[tex]ct\cdot 1=\text{ln}(\frac{d}{a})[/tex]
[tex]ct=\text{ln}(\frac{d}{a})[/tex]
Now we will divide both sides by c as:
[tex]\frac{ct}{c}=\frac{\text{ln}(\frac{d}{a})}{c}[/tex]
[tex]t=\frac{\text{ln}(\frac{d}{a})}{c}[/tex]
Therefore, our solution would be [tex]t=\frac{\text{ln}(\frac{d}{a})}{c}[/tex].
We have the equation:
[tex]a*e^{c*t} = d[/tex]
And we want to solve it for t, we will get:
[tex]t = \frac{Ln(d) - Ln(a)}{c}[/tex]
Remember the rule:
[tex]Ln(e^x) = x[/tex]
and the rule:
[tex]Ln(A*B) = ln(A) + Ln(B)[/tex]
Then we can apply the natural logarithm to both sides to get:
[tex]Ln(a*e^{c*t}) = Ln(d)[/tex]
Now we use the second rule and then the first rule to get:
[tex]Ln(a*e^{c*t}) = Ln(d)\\\\Ln(a) + Ln(e^{c*t}) = Ln(d)\\\\Ln(a) + c*t = Ln(d)\\\\t = \frac{Ln(d) - Ln(a)}{c}[/tex]
This is the solution we wanted.
If you want to learn more, you can read:
https://brainly.com/question/24626259
