Express the function f(x)=g(x)^h(x) in terms of the natural logarithmic and natural exponential functions​ (base e).

Question

f(x)=

Answer 1

To express f(x) = g(x)^h(x) in terms of the natural logarithmic and natural exponential functions (base e), we can take the natural logarithm of both sides to obtain:

ln(f(x)) = ln(g(x)^h(x))

Using the logarithmic property ln(a^b) = b * ln(a), we can rewrite the right side of the equation as:

ln(f(x)) = h(x) * ln(g(x))

Next, we can exponentiate both sides with base e:

e^(ln(f(x))) = e^(h(x) * ln(g(x)))

Since e^(ln(x)) = x, we can simplify the left side to:

f(x) = e^(h(x) * ln(g(x)))

Therefore, f(x) can be expressed as e raised to the power of h(x) multiplied by the natural logarithm of g(x).