The function $$ y(t) $$ is given by $$ y=\mathrm{e}^{t} $$ (a) Calculate values of $$ t $$ such that $$ y $$ is (i) 1 , (ii) 2 , (iii) 4 , (iv) 8 . (b) Show that if $$ t $$ is increased by $$ \ln 2 $$ then the value of $$ y $$ doubles. (c) A function $$ z(t) $$ is given by

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### Part (a)

We need to find values of $$ t $$ such that $$ y(t) = \mathrm{e}^{t} $$ equals specific values.

1. **For $$ y = 1 $$**:
   $$
   \mathrm{e}^{t} = 1
   $$
   Taking the natural logarithm of both sides:
   $$
   t = \ln(1) = 0
   $$

2. **For $$ y = 2 $$**:
   $$
   \mathrm{e}^{t} = 2
   $$
   Taking the natural logarithm:
   $$
   t = \ln(2)
   $$

3. **For $$ y = 4 $$**:
   $$
   \mathrm{e}^{t} = 4
   $$
   Taking the natural logarithm:
   $$
   t = \ln(4) = \ln(2^2) = 2\ln(2)
   $$

4. **For $$ y = 8 $$**:
   $$
   \mathrm{e}^{t} = 8
   $$
   Taking the natural logarithm:
   $$
   t = \ln(8) = \ln(2^3) = 3\ln(2)
   $$

### Summary of Part (a):
- $$ y = 1 $$ when $$ t = 0 $$
- $$ y = 2 $$ when $$ t = \ln(2) $$
- $$ y = 4 $$ when $$ t = 2\ln(2) $$
- $$ y = 8 $$ when $$ t = 3\ln(2) $$

### Part (b)

We need to show that if $$ t $$ is increased by $$ \ln 2 $$, then the value of $$ y $$ doubles.

Let $$ t' = t + \ln(2) $$. Then:
$$
y(t') = \mathrm{e}^{t'} = \mathrm{e}^{t + \ln(2)} = \mathrm{e}^{t} \cdot \mathrm{e}^{\ln(2)} = \mathrm{e}^{t} \cdot 2
$$
Thus, if $$ y(t) = \mathrm{e}^{t} $$, then:
$$
y(t') = 2y(t)
$$
This shows that increasing $$ t $$ by $$ \ln(2) $$ doubles the value of $$ y $$.

### Part (c)

It seems that the question is incomplete as it ends with "A function $$ z(t) $$ is given by". Please provide the definition of the function $$ z(t) $$ so that I can assist you further.
(a) For $$ y = 1 $$, $$ t = 0 $$; for $$ y = 2 $$, $$ t = \ln(2) $$; for $$ y = 4 $$, $$ t = 2\ln(2) $$; for $$ y = 8 $$, $$ t = 3\ln(2) $$. (b) Increasing $$ t $$ by $$ \ln(2) $$ doubles the value of $$ y $$.

The function is given by

(a) Calculate values of such that is (i) 1 , (ii) 2 , (iii) 4 , (iv) 8 .
(b) Show that if is increased by then the value of doubles.
© A function is given by

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The function is given by (a) Calculate values of such that is (i) 1 , (ii) 2 , (iii) 4 , (iv) 8 . (b) Show that if is increased by then the value of doubles. © A function is given by