Inverses of Exponential Functions

Why? $\log_{b}(b^{x})$ asks “b to what power gives $b^{x}$?” The answer is obviously $x$. And $b^{\log _{b} x}$ raises $b$ to exactly the exponent that produces $x$, so the result is $x$.

Swapping the rows of an exponential table gives a logarithm table:

Every point $(a, b)$ on $y = 2^{x}$ becomes $(b, a)$ on $y = \log_{2} x$.

Because the graphs are reflections, every feature swaps its $x$ and $y$ roles:

Reading the notation $x$ → $0^{+}$. The small $^{+}$ means $x$ approaches 0 from the RIGHT, through positive values like 0.1, 0.001, 0.000001. That's the only way to approach 0 inside the domain of log. As $x$ → $0^{+}$, $\log_{2} x$ → −∞: $\log_{2} 0.001 \approx -9.97$ and $\log_{2} 0.000001 \approx -19.93$, falling without bound. That is the vertical asymptote $x = 0$.

If $0 < b < 1$, both functions are decreasing. For example, $\log_{0.5} 4 = -2$ and $\log_{0.5}(1/4) = 2$. The logarithm graph is then concave up, the reflection of a decreasing, concave up exponential.

The method is the same as in 2.8: solve for $x$, then swap. The new step is converting between exponential form and log form (2.9).

Remember from 2.3: adding to the input of an exponential multiplies its output ($f(x + 1) = b \cdot f(x)$). The inverse reverses that: multiplying the input of a logarithm adds to its output.

Doubling the input of $\log_{2}$ always adds exactly 1. That's why a logarithm grows so slowly: $\log_{2} x$ reaches 10 only at $x = 1,024$, and 20 only after doubling ten more times.

Check with the table: $N^{-1}(864) = \log_{1.5}(3.375) = 3$, because $1.5^{3} = 3.375$. And $N^{-1}(10000) = \log_{1.5} 39.0625 \approx 9.04$, the number from 2.9. Doubling the audience from 256 to 512 takes $N^{-1}(512) = \log_{1.5} 2 \approx 1.71$ months, the doubling time estimated in 2.5A.