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## Exponential and Logarithmic Equations
We use the inverse properties to solve both exponential and logarithmic equations:
Since the bases are the same we can set the exponents equal to each other.
Here the bases are different so let's take logs of both sides. Bring the exponents down in front of the logs. Divide both sides by the constant This is the exact solution.
Multiply both sides by 2 to eliminate the fraction. We can write this as Multiply both sides by to eliminate the fraction. This is like a quadratic equation! Use the quadratic formula: Take ln of both sides
Notice that there is a solution close to x =1. If we zoom in we can get a closer approximation: x = 0.96 MathCAD won't solve it if you use Symbolics-Variable-Solve. But if you use the Given-Find block. It will solve it numerically:
This is a quadratic equation
Now raise both sides as exponents of 2 Solution is:
where t is measured in minutes. a) Find the runner's pulse at the end of the race and also 1 minute after the end of the race. b) How long, to the nearest minute, after the end of the race will the runner's pulse be at 80 beats per minute? a) beats per minute at
the end of the race b) and solve for t. Solution is:
a) In how many seconds will the velocity be 50 feet per second? b) Determine the horizontal asymptote for the graph of this function. c) Write a sentence that describes the meaning of the horizontal asymptote in the context of this problem. a) We need to solve Multiply both sides by the denominator . b) Change the function to Now as t gets large without bound, the two little fractions disappear. So the horizontal asymptote is y = 100. See the graph below. c) The maximum (terminal) velocity of the object is 100
feet per second |