Question 151·Hard·Advanced Math

Let $q(x)=ax^2+px+c$ be the exponent in $f(x)=b^{q(x)}$. Since $b>1$, each power in the table tells you its exponent: $q(1)=2$, $q(2)=7$, and $q(4)=23$. Fit the exponents, not the powers of $b$.

In a Desmos table, put $1,2,4$ in $x_1$ and the matching exponents $2,7,23$ in $y_1$. Each row is one point on $q$. The table is ready for the quadratic fit.

Type $y_1\sim ax_1^2+px_1+c$. The $\sim$ tells Desmos to run a quadratic regression on the three points. Under PARAMETERS, it reports $a=1$, $p=2$, and $c=-1$, so the exponent rule is $q(x)=x^2+2x-1$.

The condition $f(k)=b^{14}$ also means $f(k)=b^{q(k)}$. Both powers have the same base $b>1$, so their exponents must match: $q(k)=14$. Here $14$ is the output of $q$, not the input $k$.

Use the fitted rule and type $k_1^2+2k_1-1\sim14\left\{k_1>0\right\}$. Desmos fits $k_1$ to solve the equation; the braces enforce the positive-input condition because a quadratic can hit the same output twice. Under PARAMETERS, Desmos reports $k_1=3$. So the requested value of $k$ is $3$. Grid in 3.