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Ahat [919]
3 years ago
12

For the equation ae^ct=d, solve for the variable t in terms of a,c, and d. Express your answer in terms of the natural logarithm

.
Mathematics
1 answer:
saveliy_v [14]3 years ago
5 0

We have been given an equation ae^{ct}=d. We are asked to solve the equation for t.

First of all, we will divide both sides of equation by a.

\frac{ae^{ct}}{a}=\frac{d}{a}

e^{ct}=\frac{d}{a}

Now we will take natural log on both sides.

\text{ln}(e^{ct})=\text{ln}(\frac{d}{a})

Using natural log property \text{ln}(a^b)=b\cdot \text{ln}(a), we will get:

ct\cdot \text{ln}(e)=\text{ln}(\frac{d}{a})

We know that \text{ln}(e)=1, so we will get:

ct\cdot 1=\text{ln}(\frac{d}{a})

ct=\text{ln}(\frac{d}{a})

Now we will divide both sides by c as:

\frac{ct}{c}=\frac{\text{ln}(\frac{d}{a})}{c}

t=\frac{\text{ln}(\frac{d}{a})}{c}

Therefore, our solution would be t=\frac{\text{ln}(\frac{d}{a})}{c}.

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