State, true or false :
(i) If `log_(10)x = a`, then `10^(x) = a`
(ii) If `x^(y) = z`, then `y = log_(z) x`.
(iii) `log_(2) 8 = 3 and log_(8) 2 = (1)/(3)`.

Answer

To determine whether the statements are true or false, we will analyze each statement one by one.

### Statement (i):
**If `log_(10)x = a`, then `10^(x) = a`.**

**Step 1:** Recall the definition of logarithms. If `log_b(a) = c`, then it means `b^c = a`.

**Step 2:** In our case, we have `log_(10)x = a`. This means that `10^a = x`.

**Step 3:** The statement claims that `10^x = a`. However, from our analysis, we found that `10^a = x`, not `10^x = a`.

**Conclusion:** The statement is **False**.

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### Statement (ii):
**If `x^(y) = z`, then `y = log_(z) x`.**

**Step 1:** Start with the equation `x^y = z`.

**Step 2:** Taking logarithm on both sides, we have `log(z) = log(x^y)`.

**Step 3:** Using the power rule of logarithms, we can rewrite this as `log(z) = y * log(x)`.

**Step 4:** Rearranging gives us `y = log(z) / log(x)`.

**Step 5:** By the change of base formula, we can express this as `y = log(x) base z`.

**Conclusion:** The statement is **False**.

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### Statement (iii):
**`log_(2) 8 = 3 and log_(8) 2 = (1)/(3)`.**

**Step 1:** Evaluate `log_(2) 8`. We know that `8 = 2^3`, so `log_(2) 8 = log_(2) (2^3)`.

**Step 2:** Using the power rule, we find `log_(2) 8 = 3 * log_(2) 2 = 3 * 1 = 3`.

**Step 3:** Now evaluate `log_(8) 2`. We can express `8` as `2^3`, so `log_(8) 2 = log_(2^3) 2`.

**Step 4:** Using the change of base formula, we have `log_(8) 2 = log(2) / log(8) = log(2) / (3 * log(2)) = 1/3`.

**Conclusion:** Both parts of the statement are **True**.

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### Summary of Answers:
1. Statement (i): **False**
2. Statement (ii): **False**
3. Statement (iii): **True**

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