Exponential and logarithmic graphs

Build a graph by multiplying

Start with a single unit and double it at each whole step. After one step you have two, after two steps four and after three steps eight. Walking a step backwards requires division by two. This gives one half at negative one and one quarter at negative two. These coordinates belong to the exponential function (funzione esponenziale) $y=2^x$. At half a step the factor must square to two when you take the same half step again, which explains the appearance of a square root for a fractional exponent.

With any positive base $a$ other than one, the graph passes through $(0,1)$ and $(1,a)$. A positive base makes all real powers positive. When the base exceeds one each unit step multiplies the height by something greater than one, and the curve increases. For a base between zero and one the same step multiplies the height by a proper fraction. The curve then decreases. In the direction in which the heights shrink they approach zero without reaching it at any finite input. The horizontal axis is an asymptote.

Reverse the question

A logarithm (logaritmo) asks for the exponent. The statement $\log_a b=c$ says precisely that $a^c=b$. It requires a positive argument and a positive base different from one. Base one would always give one as its output, so there would be no unique exponent to recover. The value of a logarithm can be negative because a negative exponent still gives a positive power. Separating the sign of the argument from the sign of the answer prevents a common domain error.

Interchanging a point's two coordinates reflects it in the line $y=x$. Every exact point on an exponential graph gives a point on the corresponding logarithmic graph this way. The point $(0,1)$ becomes $(1,0)$, and the horizontal asymptote becomes the vertical line $x=0$. The domain of a logarithm is the positive real numbers while its range includes all real values. Reflection preserves the increasing or decreasing character. With a base of one half the smaller of two positive arguments therefore has the larger logarithm.

Translate powers into logarithm laws

Take two positive numbers $u$ and $v$ and write them as powers of the same base. Multiplying the numbers adds their exponents. Consequently $\log_a(uv)=\log_a u+\log_a v$. Dividing subtracts the exponents, and raising a positive argument to a power multiplies its logarithm by that power. Each rule follows an operation on powers. Addition of arguments has no matching product rule. Testing a proposed identity with two inputs equal to one often exposes this error immediately.

Keep the original domain when combining expressions. Two separate logarithms in a sum each require a positive argument. Their product can also be positive when both factors are negative, so a combined logarithm considered on its own may allow more inputs than the original sum. When expanding a logarithm of a square, the correct real expression for a nonzero input is $\log_a(x^2)=2\log_a|x|$. The absolute value allows both signs that the square already permitted.

Compare without estimating decimals

Try bracketing an argument by adjacent integer powers. Since nine is between eight and sixteen, its base-two logarithm lies strictly between three and four. For a decreasing exponential, put the corresponding exponents in reverse order when reading the interval. If two logarithms have different bases, the change-of-base formula $\log_a b=\log_c b/\log_c a$ can express them through one familiar base. To justify it, write $a^x=b$ and take the base-$c$ logarithm of both sides. The unknown exponent multiplies the logarithm of $a$, so division isolates it. A useful choice for the new base makes at least one of those two logarithms an integer.