Exponential and logarithmic functions

The next couple of weeks will be about exponential and logarithmic functions. Logarithms are particularly useful in many applications of mathematics.

Exponential functions

A power function is a function f of the form

f(x) = xⁿ,

for some constant n called the exponent of the function. You already know about power functions.

An exponential function is a function f of the form

f(x) = b^x,

for some constant b called the base of the function. The base should be a positive number, so that b^x makes sense for every real number x.

If you don't remember any other values of an exponential function, remember these:

f(0) = b⁰ = 1,
f(1) = b¹ = b,
f(−1) = b⁻¹ = 1/b.

The domain of an exponential function is the set of all real numbers; as long as b ≠ 1, the range is the set of all positive numbers. (Because b is positive, b^x is also positive.) If b > 1, then the exponential function is increasing; if b < 1, then the exponential function is decreasing; and if b = 1, then the exponential function is constant.

Logarithmic functions

As long as b ≠ 1, the exponential function with base b is one-to-one, so it has an inverse. A logarithmic function is an inverse of an exponential function. These two statements mean exactly the same thing:

b^x = y;
log_b y = x.

If you don't remember any other values of a logarithmic function, remember these:

log_b 1 = 0,
log_b b = 1,
log_b (1/b) = −1.

The domain of a logarithmic function is the set of all positive numbers; the range is the set of all real numbers. (A logarithm of a negative number is imaginary.) If b > 1, then the exponential function is increasing; if b < 1, then the exponential function is decreasing.

There are abbreviations for logarithms with certain special bases:

lb x = log₂ x;
lg x = log₁₀ x;
ln x = log_e x, where e is a special number, about 2.72;
log x is the logarithm of x with whatever is your favourite base.

The book's favourite base is 10, which I will also use.

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This web page was written in 2011 by Toby Bartels, last edited on 2011 May 11. Toby reserves no legal rights to it.

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