A logarithm answers an exponent question. The statement \(\log_b y=a\) means exactly the same thing as \(b^a=y\), and keeping those two forms connected makes most logarithm rules easier to understand.

The same relationship in two directions

The solid curve is y = b^x; the lighter curve is its inverse, the logarithm base b; the dashed diagonal is y = x. The two marked points are reflections of each other: if b^a = y, then the logarithm of y is a. One fact, plotted twice, as the panels on either side of the graph spell out. Slide the base through e ≈ 2.718; nothing dramatic marks it, and that is rather the point. Push the base higher and the exponential steepens while the logarithm flattens.

The exponential and logarithmic graphs are reflections across \(y=x\). If \(b^a=y\), then the point \((a,y)\) lies on the exponential graph, and the reflected point \((y,a)\) lies on the logarithmic graph because \(\log_b y=a\). This is what it means for the two functions to be inverses.

For example, \(2^5=32\) and \(\log_2 32=5\) are two forms of the same fact. When a logarithmic expression looks unfamiliar, rewriting it exponentially is often the cleanest first move.

The domain and asymptote also reverse

For \(b>0,\; b\neq1\), the exponential function \(b^x\) always produces positive outputs, so its range is \((0,\infty)\). The logarithm reverses inputs and outputs, so its domain is \((0,\infty)\). The horizontal asymptote \(y=0\) of the exponential becomes the vertical asymptote \(x=0\) of the logarithm, and the geometry and the algebra say the same thing.

Logarithm laws come from exponent laws

Suppose \(m=\log_b x\) and \(n=\log_b y\), so that \(x=b^m\) and \(y=b^n\). Since \(b^m b^n=b^{m+n}\), we get \(\log_b(xy) = \log_b x+\log_b y\). Similarly,

\[\log_b\left(\frac{x}{y}\right) = \log_b x-\log_b y \qquad\text{and}\qquad \log_b(x^k) = k\log_b x\]

These are exponent laws translated into logarithmic form. There is no corresponding rule for sums, and in general \(\log_b(x+y) \neq \log_b x+\log_b y\). Logarithms turn multiplication into addition, and they do not turn addition into anything similarly simple.

Solving exponential equations

Consider \(5\cdot3^t=40\). First isolate the exponential, \(3^t=8\), and then write the equivalent logarithmic statement, \(t=\log_3 8\). Using change of base,

\[t = \frac{\ln8}{\ln3} \approx1.893\]

The reverse process solves logarithmic equations. If \(\log_2 x=5\), then \(x=2^5=32\). The useful principle is to isolate the invertible function first, then apply its inverse.

The base \(e\)

The number \(e\approx2.718\) is the standard base for continuous exponential growth. The logarithm base \(e\) is written \(\ln x\), and it follows all the same logarithm laws. Its importance becomes greater in calculus because \(e^x\) has especially simple derivative and integral behavior. For precalculus, it is enough to treat \(e\) as a particular exponential base and \(\ln\) as its inverse.

The restrictions on logarithmic bases also make sense from invertibility. At \(b=1\), the exponential function would be \(1^x=1\), a constant function. It is not one-to-one and therefore has no inverse function.