a) Solve for x: log² + log(x-3) - log(3x-4) = log 3. b) Show that log((x+y)/3) = 1/2 log x + 1/2 log y, given x+y = 7. c) Solve using laws of logarithm: i) log(99) ii) log(5) iii)... a) Solve for x: log² + log(x-3) - log(3x-4) = log 3. b) Show that log((x+y)/3) = 1/2 log x + 1/2 log y, given x+y = 7. c) Solve using laws of logarithm: i) log(99) ii) log(5) iii) log(√x) iv) log(x/59). d) Find the domain and range of: i) f(x) = x² ii) f(x) = x iii) f(x) = 2x + 3 iv) P(x) = sin x. Read more

Steps to Solve

1. Solve for x in the logarithmic equation We start with the equation:

   $$ \log^2 + \log(x-3) - \log(3x-4) = \log 3 $$

   Let's isolate $\log(x-3)$:

   $$ \log^2 + \log(x-3) = \log(3x-4) + \log 3 $$

   We can rewrite this using the properties of logarithms:

   $$ \log^2 + \log(x-3) = \log(3(3x-4)) $$

   Then, combine the logs:

   $$ \log^2 + \log(x-3) = \log(9x - 12) $$

   Now we can exponentiate both sides to eliminate the logarithm.

2. Demonstrate the logarithmic identity We are to show:

   $$ \log \left(\frac{x+y}{3}\right) = \frac{1}{2} \log x + \frac{1}{2} \log y $$

   We can start by rewriting the right side using properties of logarithms:

   $$ \frac{1}{2} \log x + \frac{1}{2} \log y = \log(x^{1/2}) + \log(y^{1/2}) = \log(\sqrt{x} \sqrt{y}) $$

   Hence, we get:

   $$ \log \left(\frac{x+y}{3}\right) = \log(\sqrt{xy}) $$

   This shows that if $x+y=2\sqrt{xy}$, the equality holds.

3. Use laws of logarithms for various problems For each logarithmic expression, we will simplify:

   i) For $\log(99)$, we can leave it as it is or express it using base change.

   ii) For $\log(\frac{x}{5})$, use:

   $$ \log(x) - \log(5) $$

   iii) For $\log(\sqrt{x})$, use:

   $$ \frac{1}{2} \log(x) $$

   iv) For $\log(\frac{x}{59})$, use:

   $$ \log(x) - \log(59) $$

4. Determine the domain and range of functions For the functions given:

   i) $f(x) = x^2$

   • Domain: All real numbers $(-\infty, \infty)$
   • Range: $[0, \infty)$

   ii) $g(x) = x^2$ is the same as above.

   iii) $h(x) = 2x + 3$

   • Domain: All real numbers $(-\infty, \infty)$
   • Range: All real numbers $(-\infty, \infty)$

   iv) $k(x) = \sin(x)$

   • Domain: All real numbers $(-\infty, \infty)$
   • Range: $[-1, 1]$