On exercise 5-6....Estimate the values of b for the two exponential functions on the above graph. At what points will each of the logarithmic functions reach the values y=2, y=3 and y=4?...I am really not sure what you mean. There are several lines on the graph and I dont know which to look at and do I just write down when y= 2 then x=.5?....Thanks
Since the table for y = b^x will contain the points (0, 1) and (1, b), since b^0 = 1 and b^1 = b, the graph of log{base b}(x) will contain the 'reversed' points (1, 0) and (b, 1).
Look at the two exponential-function graphs. You will see that they both go through (0, 1), as the exponential function y = b^x must do.
Each function also goes through the point (1, b). What are the coordinates of the x = 1 point for each graph? Those coordinates will be your values of b.
To find where the logarithmic function y = log{base b}(x) takes value 2, you have to solve the equation 2 = log{base b}(x).
One way to do solve the equation is to remember that since y = log{base b}(x) is inverse to the b^x function, y = log{base b}(x) is the same as x = b^y.
The other way is to simply raise b to the power of each side, getting the equation
b^2 = b^( log{base b}(x) )
and using the basic property of logs to see that the right-hand side is just x. The left-hand side is easy to find if you know b.
A logarithmic equation of the form log{base b} (x) = y can be inverted to exponential form by reversing the roles of x and y to get x = b^y.
That is, the equations
log{base b}(x) = y
and
b^y = x
are equivalent.
So for example if you want to find the value of x for which
log{base 7} ( x ) = 3,
you would reverse the equation to exponential form
7^3 = x
and just cube the 7 to get
x = 343.