graphing, logarithms, etc.
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1). Without using a graph utility, sketch the graph of f(x) =3^x-5
2) A certain population grows according to the equation Y=40e^0.025t. Find the initial population (to the nearest integer) when t=50.
3) Write in exponential from: logb 37=2.
4) Write as sum, difference, or multiple of logarithms: in 5x/3â??X^2+1
5) Write as the logarithm of single quantity: 1/5[3 log(x+1)+2 log(x-1)-log7].
6) Solve x: 2x+in e^4x=12.
7) Solve x: log3(x^2+5)=log3(4x^2-2x).
8) Determine the annual rate of interest compounded continuously for the sum of money in an account to become triple the original amount in 10 years.
9) The spread of flu virus through a certain population is modeled by:
Y=1000/1+990e^-0.7t. Where y is the total number infected after t days. In how many days will 612 people be infected with the virus?
10) Evaluate logA 9/2, given logA 2=0.2789, logA 3=0.4421.
11) Find the equation of the inverse of f(x)=5x+2. Graph f and f^-1 on the same set of axes.
12) Show that if f(x) = 4x-1, then f^-1(x) = x+1/4
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Solution Summary
This solution uses the properties of logarithms to simplify the logarithm expression into a single log.
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Please see the attached solutions.
1. Since I can't really sketch it for you, you can just plug in a few values for x, find the corresponding y values, and plot the points. The point of this problem, I think, is to test for your knowledge of shifts. Basically, the graph should look like a steeper than normal exponential curve shifted downwards by 5.
2. Here, we can just plug 50 into the equation for t and solve. Y = 40e^(0.025*50) = 40e^1.25 = 40*3.490 = 139.614. Since they ask to the nearest integer, we give 140 as the answer.
3. logb37 = 2 written in exponential form would be b^2 = 37
4. I'm not sure what you meant to write here, since the characters are all messed up. If you add a comment with the correction, I'll gladly solve it for you.
5. For this problem, we use the ...
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