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# Logarithmic identities

What follows is a list of identities that are useful when dealing with logarithms. All of these are valid for all positive real numbers a, b and c except that the base of a logarithm may never be 1.

 Table of contents 1 Special values 2 Multiplication, division and exponentiation 3 Logarithms and exponential functions are inverses 4 Change of base formula 5 Limits 6 Derivative 7 Integral

loga(1) = 0

loga(a) = 1

### Multiplication, division and exponentiation

logc(ab) = logca + logcb

logc(a/b) = logca - logcb

logc(ar) = r logc(a)     for all real numbers r

These three identities lead to the use of logarithm tables and slide rules; knowing the logarithm of two numbers allows you to multiply and divide them quickly, as well as compute powers and roots.

### Logarithms and exponential functions are inverses

aloga(b) = b

loga (ar) = r     for all real numbers r

These are used to solve equations in which the unknowns occur in the exponent.

### Change of base formula

logab = (logcb)/(logca)

This identity is needed to evaluate logarithms on calculators. For instance, most calculators have buttons for ln and for log10, but not for log2. To find log2(100), you have to calculate log10(100) / log10(2) (or ln(100)/ln(2), which is the same thing).

### Limits

limx->0 loga(x) = -∞     if a > 1

limx->0 loga(x) = ∞     if a < 1

limx-> loga(x) = ∞     if a > 1

limx-> loga(x) = -∞     if a < 1

limx->0 loga(x) * xb = 0

limx-> loga(x) / xb = 0

The last limit is often summarized as "logarithms grow slower than any power or root of x".

### Derivative

d/dx loga(x) = 1 / (x ln(a))

### Integral

∫ loga(x) = x loga(x) - loga(x)