(a) Given that $f(x) = 6x - 5$ and $g(x) = \frac{x + 5}{6}$, investigate if $f(g(x)) = g(f(x))$ - Leaving Cert Mathematics - Question 3 - 2020

To rewrite the equation $y = 5x^2$ in the form $ext{log}_5 y = a + b ext{log}_5 x$:

1. Taking the logarithm of both sides gives: $ext{log}_5 y = ext{log}_5 (5x^2).$

2. Using properties of logarithms, this can be written as: $ext{log}_5 y = ext{log}_5 5 + ext{log}_5 (x^2) = 1 + 2 \text{log}_5 x.$

From this, we can identify:

• $a = 1$
• $b = 2$.

To solve the equation:

1. Rewrite it using properties of logarithms: $\text{log}_5 y = \text{log}_5 \left( 5^2 \left( \frac{126}{25} x - 1 \right) \right).$

2. This implies: $y = 25 \left( \frac{126}{25} x - 1 \right).$
   Thus, $y = 126x - 25.$

3. For $y$ to be valid, note that rac{126}{25} x - 1 > 0, leading to: $x > \frac{25}{126}.$

4. Therefore, the real values of $y$ meeting this condition are: $y = 126x - 25, \quad \text{for } x > \frac{25}{126}.$