Understanding Logarithms

Question

I’m currently in my senior year of high school and we just started on the topic of logs, when doing textbook work I encountered a problem and I am confused on where I’m going wrong. Could any body help?

$$2^x+1 = 3^x-1 \implies x\log2 + \log2 = x\log3 - \log3$$ After this I don’t know where to go.

Answer 1

Remember, the values $\log 2$ and $\log 3$ are just real numbers, so you're now trying to solve the equation $ax + a = bx - b$, where $a=\log 2$ and $b = \log 3$.

Answer 2

It is $$x(\ln(2)-\ln(3))=-\ln(2)-\ln(3)$$

Answer 3

HINT

$$\ldots \iff x\log3-x\log2 =\log3+\log2 $$

Answer 4

Taking the log of the LHS and RHS is \begin{align*} \log(2^{x + 1}) &= \log(3^{x-1})\\ (x+1)\cdot\log(2) &= (x-1)\cdot\log(3)\\ x\log(2) + \log(2) &= x\log(3) - \log(3)\\ \Rightarrow x &= -\frac{\log(3) + \log(2)}{\log(2) - \log(3)} \approx 42.78468556 \end{align*}