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This question may also be related to how certain functions behave under functions of their variables. In this context, the property of commuting with binary operators, such as addition and multiplication, can be used to define classes of functions:

1. additive commutation: if $g(x, y) = x + y$, then $f\big(g(x, y)\big) = g\big(f(x),\ f(y)\big)$ if and only if $f(x + y) = f(x) + f(y)$ thus $f$ is a homogeneous linear function of the form $f(x; a) \equiv ax$

2. multiplicative commutation: if $g(x, y) = xy$, then $f\big( g(x, y) \big) = g\big(f(x),\ f(y)\big)$ if and only if $f(xy) = f(x)f(y)$ thus $f$ is "scale invariant" i.e. a power law of the form $f(x; a) \equiv x^a$

3. log-additive commutation: if $g(x, y) = x + y$, then $\log f\big( g(x, y) \big) = g\big( \log f(x),\ \log f(y) \big)$ if and only if $f(x + y) = f(x)f(y)$ thus $f$ is an exponential function of the form $f(x; a) \equiv \exp(ax)$

The last item (3) involves a third function (the logarithm) which when denoted as $h$ gives

$h\big(f[g(x, y)]\big) = g\big(h[f(x)],\ h[f(y)]\big)$

or

$h \circ f \circ g(x, y) = g\big(h \circ f(x),\ h \circ f(y)\big).$

Since $h \circ f$ occurs on both sides, we can denote this as $\tilde f$ to get

$\tilde f \big( g(x, y) \big) = g \big( \tilde f(x), \tilde f(y) \big).$

This question may also be related to how certain functions behave under functions of their variables. In this context, the property of commuting with binary operators, such as addition and multiplication, can be used to define classes of functions:

1. additive commutation: if $g(x, y) = x + y$, then $f\big(g(x, y)\big) = g\big(f(x),\ f(y)\big)$ if and only if $f(x + y) = f(x) + f(y)$ thus $f$ is a homogeneous linear function of the form $f(x; a) \equiv ax$

2. multiplicative commutation: if $g(x, y) = xy$, then $f\big( g(x, y) \big) = g\big(f(x),\ f(y)\big)$ if and only if $f(xy) = f(x)f(y)$ thus $f$ is "scale invariant" i.e. a power law of the form $f(x; a) \equiv x^a$

3. log-additive commutation: if $g(x, y) = x + y$, then $\log f\big( g(x, y) \big) = g\big( \log f(x),\ \log f(y) \big)$ if and only if $f(x + y) = f(x)f(y)$ thus $f$ is an exponential function of the form $f(x; a) \equiv \exp(ax)$

The last item (3) involves a third function (the logarithm) which when denoted as $h$ gives

$h\big(f[g(x, y)]\big) = g\big(h[f(x)],\ h[f(y)]\big)$

or

$h \circ f \circ g(x, y) = g\big(h \circ f(x),\ h \circ f(y)\big).$

Since $h \circ f$ occurs on both sides, we can denote this as $\tilde f$ to get

$\tilde f \big( g(x, y) \big) = g \big( \tilde f(x), \tilde f(y) \big)$

which has the same form as item (1) above. From this perspective, items (1) and (3) above can be seen as being isomorphic under the $\exp$ and $\log$ pair of invertible mappings.

This question may also be related to how certain functions behave under functions of their variables. In this context, the property of commuting with binary operators, such as addition and multiplication, can be used to define classes of functions:

1. additive commutation: if $g(x, y) = x + y$, then $f\big(g(x, y)\big) = g\big(f(x),\ f(y)\big)$ if and only if $f(x + y) = f(x) + f(y)$ then $f$ is a homogeneous linear function of the form $f(x; a) \equiv ax$

2. multiplicative commutation: if $g(x, y) = xy$, then $f[g(x, y)] = g[f(x),\ f(y)]$ if and only if $f(xy) = f(x)f(y)$ then $f$ is "scale invariant" i.e. a power law of the form $f(x; a) \equiv x^a$

3. log-additive commutation: if $g(x, y) = x + y$, then $\log f[g(x, y)] = g[\log f(x),\ \log f(y)]$ if and only if $f(x + y) = f(x)f(y)$ then $f$ is an exponential function of the form $f(x; a) \equiv \exp(ax)$

The last item (3) involves a third function (the logarithm) which when denoted as $h$ gives

$h\big(f[g(x, y)]\big) = g\big(h[f(x)],\ h[f(y)]\big)$

or

$h \circ f \circ g(x, y) = g\big(h \circ f(x),\ h \circ f(y)\big).$

Since $h \circ f$ occurs on both sides, we can denote this as $\tilde f$ to get

$\tilde f \big( g(x, y) \big) = g \big( \tilde f(x), \tilde f(y) \big)$

This question may also be related to how certain functions behave under functions of their variables. In this context, the property of commuting with binary operators, such as addition and multiplication, can be used to define classes of functions:

1. additive commutation: if $g(x, y) = x + y$, then $f\big(g(x, y)\big) = g\big(f(x),\ f(y)\big)$ if and only if $f(x + y) = f(x) + f(y)$ thus $f$ is a homogeneous linear function of the form $f(x; a) \equiv ax$

2. multiplicative commutation: if $g(x, y) = xy$, then $f\big( g(x, y) \big) = g\big(f(x),\ f(y)\big)$ if and only if $f(xy) = f(x)f(y)$ thus $f$ is "scale invariant" i.e. a power law of the form $f(x; a) \equiv x^a$

3. log-additive commutation: if $g(x, y) = x + y$, then $\log f\big( g(x, y) \big) = g\big( \log f(x),\ \log f(y) \big)$ if and only if $f(x + y) = f(x)f(y)$ thus $f$ is an exponential function of the form $f(x; a) \equiv \exp(ax)$

The last item (3) involves a third function (the logarithm) which when denoted as $h$ gives

$h\big(f[g(x, y)]\big) = g\big(h[f(x)],\ h[f(y)]\big)$

or

$h \circ f \circ g(x, y) = g\big(h \circ f(x),\ h \circ f(y)\big).$

Since $h \circ f$ occurs on both sides, we can denote this as $\tilde f$ to get

$\tilde f \big( g(x, y) \big) = g \big( \tilde f(x), \tilde f(y) \big).$

This question may also be related to how certain functions behave under functions of their variables. In this context, the property of commuting with binary operators, such as addition and multiplication, can be used to define classes of functions:

1. additive commutation: if g(x, y) = x + y, then f(g(x, y)) = g(f(x), f(y)) if and only if f(x + y) = f(x) + f(y) => f is a homogeneous linear function of the form f(x; a) := ax

2. multiplicative commutation: if g(x, y) = xy, then f(g(x, y)) = g(f(x), f(y)) if and only if f(xy) = f(x)f(y) => f is "scale invariant" i.e. a power law of the form f(x; a) := x^a

3. log-additive commutation: if g(x, y) = x + y, then log f(g(x, y)) = g(log f(x), log f(y)) if and only if f(x + y) = f(x)f(y) => f is an exponential function of the form f(x; a) := exp(ax)

This question may also be related to how certain functions behave under functions of their variables. In this context, the property of commuting with binary operators, such as addition and multiplication, can be used to define classes of functions:

1. additive commutation: if $g(x, y) = x + y$, then $f\big(g(x, y)\big) = g\big(f(x),\ f(y)\big)$ if and only if $f(x + y) = f(x) + f(y)$ then $f$ is a homogeneous linear function of the form $f(x; a) \equiv ax$

2. multiplicative commutation: if $g(x, y) = xy$, then $f[g(x, y)] = g[f(x),\ f(y)]$ if and only if $f(xy) = f(x)f(y)$ then $f$ is "scale invariant" i.e. a power law of the form $f(x; a) \equiv x^a$

3. log-additive commutation: if $g(x, y) = x + y$, then $\log f[g(x, y)] = g[\log f(x),\ \log f(y)]$ if and only if $f(x + y) = f(x)f(y)$ then $f$ is an exponential function of the form $f(x; a) \equiv \exp(ax)$

The last item (3) involves a third function (the logarithm) which when denoted as $h$ gives

$h\big(f[g(x, y)]\big) = g\big(h[f(x)],\ h[f(y)]\big)$

or

$h \circ f \circ g(x, y) = g\big(h \circ f(x),\ h \circ f(y)\big).$

Since $h \circ f$ occurs on both sides, we can denote this as $\tilde f$ to get

$\tilde f \big( g(x, y) \big) = g \big( \tilde f(x), \tilde f(y) \big)$