We want to solve the following equation.
log_2x=2^x
Since the method of solving the equation is not specified, we can solve it by graphing. The left-hand side of the equation represents a logarithmic function and the right-hand side represents an exponential function. Recall that logarithmic functions are inverses of exponential functions. Moreover, they have the same bases.
log_2x= 2^x
This means that the function on the left-hand side is the inverse of the function on the right-hand side. For further explanation, see the end of this solution. Therefore, their graphs are symmetric to each other with respect to the line y=x. Let's first graph y=2^x. To do so, we can make a table of values.
x
2^x
y=2^x
- 2
2^(- 2)
1/4
- 1
2^(- 1)
1/2
0
2^0
1
1
2^1
2
2
2^2
4
Once we know the coordinates, let's plot and connect the points with a smooth curve.
Now, to graph y=log_2x we should reflect the obtained graph across the line y=x. We can do it by interchanging the x- and y-coordinates of the points that are on the graph.
Points
Reflection across y=x
( -2, 1/4)
( 1/4, -2)
( -1, 1/2)
( 1/2, -1)
( 0, 1)
( 1, 0)
( 1, 2)
( 2, 1)
( 2, 4)
( 4, 2)
Again, let's plot the points and connect them with a curve.
We can see that the graphs of y=2^x and y=log_2(x) do not intersect. This means that there is no real solution to the original equation.
Extra
Inverse Function
We stated that the function y=log_2x is the inverse of y=2^x. Now we will find it algebraically. To do so we should exchange the variables in y=log_2x and then solve for y.
ccc
Given Function & & Inverse Function y=log_2 x& & x=log_2 y
To isolate y in the inverse equation, we can use the definition of a logarithm.
Definition:& n=log_b m &&⇔ b^n= m
Equation:& x=log_2 y &&⇔ 2^x= y
We found that y=2^x. Therefore, y=log_2x is the inverse of y=2^x.
