SOLUTION: I would like to get some assistance with this question. Give an example of an exponential function. Convert this exponential function to a logarithmic function. Plot the graph. t

Question 172962: I would like to get some assistance with this question. Give an example of an exponential function. Convert this exponential function to a logarithmic function. Plot the graph.
thank you
Found 2 solutions by solver91311, gonzo:
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
is an exponential function. Notice that the independent variable is an exponent.

The inverse function is

Selecting b = 10, the red line is the graph of and the green line is the graph of

Note that there are imperfections in the rendering of these graphs. The exponential function is asymptotic to the negative x-axis, and the log function is asymptotic to the negative y-axis.

Answer by gonzo(654) (Show Source): You can put this solution on YOUR website!
the general form of an exponential function is:

where:
b is the base
and
x is the exponent
and
y is the result of the base raised to the exponent.
-----
the general form of a logarithmic function is:

where:
b is the base
and
y is the exponent
and
x is the result of the base raised to the exponent.
-----
the rule for logarithms is:
if and only if:
conversely:
if and only if:
-----
the exponential form of the equation and the logarithmic form of the equation are inverses of each other.
here's an example:
10^2 = 100
-----
the exponential form of this equation is:

x = 2
y = 100
coordinate point on the graph of this equation is: (2,100)
-----
the logarithmic form of this equation is:

x = 100
y = 2
coordinate point on the graph of this equation is: (100,2)
----
in general:
if f(x) = (a,b), and g(x) = (b,a) then the equations are inverses of each other.
this is true for the two equations:

and

-----
you can also graph these functions (i used the base of 10 - any base would work).
the functions are:

and

-----
we used base b = 10 so that it can be graphed easily and solved for easily on the calculator. the base can be any real number > 0.
-----
the line y = x is inserted in the middle of the graph to show more clearly that these functions are inverses of each other which shows up as their being reflections about the line y = x (kind of like mirror images if you think of the line y = x as the mirror).
-----
look below the graph for further comments.
-----

-----
a practical example:
suppose you are given:

how would you solve this?
let y = 4500
let x = x
let b (the base) = 10
using the laws of logarithms, you know that:

if and only if:

so you go to the calculator and get the log of 4500 which is: 3.653212514
how do you know this is good?
you use the calculator again to take to get: 4500
-----
what if the base is not 10?
how to solve?
there's a conversion formula you can use to convert any base to the base of 10 so you can solve it on your calculator.
that conversion formula is:

-----
here's an example of how that works.
-----
your new problem:

you know that this is true if and only if:

you can't solve that on your calculator directly, but you can convert it to the base of 10 using the conversion formula.
that would be:

-----
you use your calculator to get:

which becomes:
x = 8.341946099
this means that:

you can use your calculator to prove that this is true:
34561 = 34561 proving that it is true.
-----

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