Question

Solve the following equations

In(x-2) +In2 = Iny

Log₃x = y = Log₉(2x-1)

Answer 1

Explanations & Calculations

• The logarithmic notation can be taken out and a linear equation can be written between x and y.

$\qquad\qquad \begin{aligned} \small \ln{(x-2)}+\ln 2&=\small \ln y\\ \small \ln(x-2)&=\small \ln y-\ln 2\\ \small \ln(x-2)&=\small \ln{\frac{y}{2}}\\ \small (x-2)&=\small \frac{y}{2}\\ \small y&=\small 2x-4\cdots(1) \end{aligned}$

• A relationship between $\small x$  & $\small y$ and also the value of $\small x$ directly can be found by the second set of equations as it contains 2 equations in one line.

$\qquad\qquad \begin{aligned} \small \log_3(x)&=\small y\\ \small x&=\small 3^y\cdots\cdots(2)\\ \\ \small \log_3(x)&=\small \log_9(2x-1)\\ &=\small \frac{\log_3(2x-1)}{\log_3(9)}=\frac{\log_3(2x-1)}{2}\\ \small 2\log_3(x)&=\small\log_3(2x-1)\\ \small \log_3(x^2)&=\small \log_3(2x-1)\\ \small x^2&=\small 2x-1\to x^2-2x+1=0\to(x-1)^2=0\\ \small x &=\small \begin{cases} 1-\text{twice} \end{cases} \end{aligned}$

• Then the one of the set of corresponding values of $\small y$ can be calculated through $\small (1)$ and the others through $\small (2)$ .