If $$ f(x)=14^{x} $$, what is $$ f^{-1}(x) ? $$ $$ f^{-1}(x)=\square $$

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To find the inverse function $$ f^{-1}(x) $$ of the function $$ f(x) = 14^x $$, we start by setting $$ y = f(x) $$:

$$
y = 14^x
$$

Next, we want to solve for $$ x $$ in terms of $$ y $$. To do this, we can take the logarithm of both sides. Using the natural logarithm (though any logarithm would work), we have:

$$
\ln(y) = \ln(14^x)
$$

Using the property of logarithms that states $$ \ln(a^b) = b \ln(a) $$, we can rewrite the right side:

$$
\ln(y) = x \ln(14)
$$

Now, we can solve for $$ x $$:

$$
x = \frac{\ln(y)}{\ln(14)}
$$

Since we are looking for the inverse function $$ f^{-1}(x) $$, we replace $$ y $$ with $$ x $$:

$$
f^{-1}(x) = \frac{\ln(x)}{\ln(14)}
$$

Thus, the inverse function is:

$$
\boxed{\frac{\ln(x)}{\ln(14)}}
$$
$$ f^{-1}(x) = \frac{\ln(x)}{\ln(14)} $$

If \( f(x)=14^{x} \), what is \( f^{-1}(x) ? \) \( f^{-1}(x)=\square \)

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