We may define a cardinal logarithm $\log_μ(κ) := \min\{λ: μ^λ ≥ κ\}$. We may also define its augmented variant: $\log^+_μ(κ) := \min\{λ: μ^λ > κ\} = \max\{λ: μ^{<λ} ≤ κ\} ≤ κ$. Hence, we have $μ^λ = κ$ if and only if $λ ∈ [\log_μ(κ), \log^+_μ(κ))$. Also observe that $\log_μ(κ) = \log_2(κ) =: \log(κ)$ for every $2 ≤ μ ≤ κ$, and analogously for $\log^+$.
My question is, what can we tell about $\log^+(\mathfrak{c})$? Clearly, it is inclusively between $ω_1$ and $\mathfrak{c}$. So I wonder, is it related to standard cardinal characteristics of the continuum? Does it have a standard name?