Exponentials and Logarithms? (see questions below)
The graph of #y=ab^x# passes through the points #(2,400)# and #(5,50)# .
a) find the values of the constants a and b
b) given that #ab^x < k# , for some constant #k > 0# , show that #x > log(1600/k)/log(2)# where log means log to any valid base
The graph of
a) find the values of the constants a and b
b) given that
2 Answers
Shown below...
Explanation:
The equation
Substituting we get..
Manipulating the first equation:
Substituting into the second equation...
Substituting into first equation...
If
Take logs:
Assuming you know your log laws!
Part a
Given:
Use the point
Use the point the point
Divide equation [1] by equation [2]:
Please observe that
We know that
Substitute the value of b into equation [1] to find the value of a:
Part b
Given:
Use the general logarithm of any base on both sides:
Use the property of logarithms that allows one to expand the product into a sum:
Multiply both sides by -1:
Add
Use the property of logarithms that says that the difference of two logarithms is the same as division.
The negative of a logarithm is the same as inverting the argument:
Use the property of logarithms that allows one to bring the exponent outside as a coefficient:
Divide both sides by