Which point satisfies both $f(x)=2^x$ and $g(x)=3^x$ ?

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Answer 1 · 2016-11-17T07:24:36

$(0, 1)$

If $f(x) = y = g(x)$ then we have:

$2^x = 3^x$

Divide both sides by $2^x$ to get:

$1 = 3^x/2^x = (3/2)^x$

Any non-zero number raised to the power $0$ is equal to $1$ . Hence $x=0$ is a solution, resulting in:

$f(0) = g(0) = 1$

So the point $(0, 1)$ satisfies $y = f(x)$ and $y = g(x)$

Note also that since $3/2 > 1$ , the function $(3/2)^x$ is strictly monotonically increasing, so $x=0$ is the only value for which $(3/2)^x = 1$

Which point satisfies both f(x)=2^xf(x)=2^x and g(x)=3^xg(x)=3^x?