4.8 out of 5 stars 188. Get it as soon as Sat, Oct 17. FREE Shipping by Amazon. More Buying Choices $57.83 (20 new offers) Related searches. Signals and Systems: Part II / Solutions S3-3 x(- t) and x(1 - t) are as shown in Figures S3.4-2 and S3.4-3. X (-t) -12 Figure S3.4-2 x(1-t) x1-0.
We next want to be able to use our calculator to evaluate a logarithm of any base. Since our calculator can only evaluate bases e and 10, we want to be able to change the base to one of these when needed. The formula below is what we need to accomplish this task.
Change of Base Formula
Proof
We write
y = loga x
So that
ay = x
Take logb of both sides we get
logb ay = logb x
Using the power rule:
y logb a = logb x
Dividing by logb a
logb x y = logb a
Example
Find
log2 7
We have
log 7 log2 7 = = 2.807.. log 2
Log Equations
Example
Solve
log2 x - log2 (x - 2) - 3 = 0
We use the following step by step procedure:
Step 1: bring all the logs on the same side of the equation and everything else on the other side.
log2 x - log2(x - 2) = 3
Step 2:Use the log rules to contract to one log
x log2 = 3 x - 2
Step 3:Exponentiate to cancel the log (run the hook).
x = 23 = 8 x - 2
Step 4:Solve for x
x = 8(x - 2) = 8x - 16
7x = 16
16 x = 7
Step 5:Check your answer
log2 (16/7) - log2 (16/7 - 2) = 3
Exercises:
log(x + 2) - log(x - 1) = 1
log2(x) + log2(x + 5) = 2
Exponential Equations
Example
Solve for x in
2x - 1 = 3x + 1
Step 1: Take logs of both sides using one of the given bases