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A logarithmic function is the inverse of an exponential function. It is used to find the exponent to which a base number must be raised to produce a given value. The general form of a logarithmic function is: f(x)=logb​(x)

Common logarithms are logarithms with a base of 10

-not necessary to write the base

-logx is the same as log10x

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An exponential function is a function in the form f(x) = bx, where “x” is a variable and “b” is a constant which is called the base of the function such that b ≠0, 1

First differences are related by multiplication

when |b|>1, exponential functions increasing (growth)

when |b|<1, exponential functions decreasing (decay)

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Power law: logamn = n · logam

where a > 0, a ≠ 0, m > 0, n ∈ R

Change of base formula: Lognm = log m/log n

where n > 0, n ≠ 1, m > 0

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y = a log [ k (x-d) ] + c

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Similarities:

The graphs of exponential functions and logarithmic functions are all curves

As we move from left to right, the graphs rising

Differences:

Exponential functions

hor. asymptote: y=0

ver. asymptote: N/A

Domain: { x ∈ R }

Range: { y > 0| y∈ R }

y-intercept: (0,1)

x-intercept: N/A

Logarithmic functions

hor. asymptote: N/A

ver. asymptote: x=0

Domain: { x> 0|x ∈ R }

Range: { y∈ R }

y-intercept: N/A

x-intercept: (1,0)

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Simplify and state any restrictions on the variables.

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Simplify and state any restrictions on the variables.