How do I solve simultaneous exponential equations without logarithm

Question

I had a question from my book

$$(3^x+1)-(2^y+1)=1$$

$$4\cdot(3^x)+3\cdot(2^y)=24$$

So without using logarithms, how can I solve it thx

Answer 1

I suspect you mean $3^{x+1}$ and $2^{y+1}$. Otherwise the equations will be impossible to solve without using logarithms. Following the assumption that you meant $3^{x+1}$ and $2^{y+1}$,

Let $3^x$ be $a$ and $2^y$ be $b$.

Substitute these in accordingly and we get:

$3a−2b=1$

$4a+3b=24$

Solving simultaneously:

$a=3$ and $b=4$

From there, we simplify the expressions of $a$ and $b$ respectively, and we obtain

$3^x=3^1$ and $2^y=2^2$

We can then derive that $x=1$ and $y=2$.