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

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

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Answer 1 · 2016-11-17T07:24:36

$(0, 1)$ $(0, 1)$

Explanation:

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

$2^{x} = 3^{x}$ $2^x = 3^x$

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

$1 = \frac{3^{x}}{2^{x}} = {(\frac{3}{2})}^{x}$ $1 = 3^x/2^x = (3/2)^x$

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

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

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

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

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

1 Answer

Explanation:

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

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