Logarithm as Inverse of Exponential

The logarithm "undoes" the exponential

$\log_b(x)$ answers: "What power of $b$ gives $x$?" It's the inverse of $b^x$:

$y = b^x \iff x = \log_b(y)$

Key relationships

• $\log_b(b) = 1$ (what power of $b$ gives $b$? Answer: 1)
• $\log_b(1) = 0$ (what power of $b$ gives 1? Answer: 0)
• $b^{\log_b(x)} = x$ (they cancel — inverse functions)
• $\log_b(b^x) = x$ (same cancellation)

Graphical relationship

The graphs of $y = b^x$ and $y = \log_b(x)$ are mirror images across the line $y = x$. Every point $(a, b)$ on the exponential corresponds to $(b, a)$ on the logarithm.

Key features of $y = \log_b(x)$

• Domain: $x > 0$ only (can't take log of zero or negative)
• x-intercept: $(1, 0)$ because $\log_b(1) = 0$
• Vertical asymptote: $x = 0$ (the y-axis)
• Passes through: $(b, 1)$ because $\log_b(b) = 1$

Try it in the visualization

Both $y = b^x$ and $y = \log_b(x)$ are graphed with the $y = x$ mirror line. Corresponding points are connected by dashed lines showing the $(a,b) \leftrightarrow (b,a)$ reflection. Adjust $b$ to see different bases. Graphically, the log curve is the exponential curve reflected across the line $y = x$.