SOLUTION: 9. Evaluate the following: a. log1/2 4 b. 1og8 64 c. log32 8 d. 1og3 81 e. pi^(log_pi 2pi)

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: 9. Evaluate the following: a. log1/2 4 b. 1og8 64 c. log32 8 d. 1og3 81 e. pi^(log_pi 2pi)      Log On


   

Question 113827: 9. Evaluate the following:
a. log1/2 4
b. 1og8 64
c. log32 8
d. 1og3 81
e. pi^(log_pi 2pi)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Remember if we have log%28b%2C%28x%29%29=y, we can write it as b%5Ey=x


a)
log%281%2F2%2C%284%29%29=y Start with the given equation. Just set the original expression equal to y. Note: the entire 1%2F2 is the base of the log.


%281%2F2%29%5Ey=4 Rewrite the original expression using the property I listed above



%282%5E%28-1%29%29%5Ey=4 Rewrite 1%2F2 as 2%5E%28-1%29


2%5E%28-y%29=4 Multiply the exponents



2%5E%28-y%29=2%5E2 Rewrite 4 as 2%5E2


Since the bases are equal, the exponents are equal. So -y=2 which means y=-2

So log%281%2F2%2C%284%29%29=-2


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b)
log%288%2C%2864%29%29=y Start with the given equation. Just set the original expression equal to y


8%5Ey=64 Rewrite the original expression using the property: log%28b%2C%28x%29%29=y <===> b%5Ey=x


8%5Ey=8%5E2 Rewrite 64 as 8%5E2


Since the bases are equal, the exponents are equal. So y=2

So log%288%2C%2864%29%29=2


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c)

log%2832%2C%288%29%29=y Start with the given equation. Just set the original expression equal to y


32%5Ey=8 Rewrite the original expression using the property: log%28b%2C%28x%29%29=y <===> b%5Ey=x


%282%5E5%29%5Ey=2%5E3 Rewrite 8 as 2%5E3 and 32 as 2%5E5


2%5E%285y%29=2%5E3 Multiply the exponents

Since the bases are equal, the exponents are equal. So 5y=3 which means y=3%2F5

So log%2832%2C%288%29%29=3%2F5

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d)

log%283%2C%2881%29%29=y Start with the given equation. Just set the original expression equal to y


3%5Ey=81 Rewrite the original expression using the property: log%28b%2C%28x%29%29=y <===> b%5Ey=x


3%5Ey=3%5E4 Rewrite 81 as 3%5E4

Since the bases are equal, the exponents are equal. So y=4

So log%283%2C%2881%29%29=4

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e)

pi%5E%28log%28pi%2C%282pi%29%29%29 Start with the given expression


pi%5E%28log%28pi%2C%282%29%29%2Blog%28pi%2C%28pi%29%29%29 Break up the logarithm using the identity log%28b%2C%28x%2Ay%29%29=log%28b%2C%28x%29%29%2Blog%28b%2C%28y%29%29. Think of 2pi as 2 times pi.


pi%5E%28log%28pi%2C%282%29%29%29%2Api%5E%28log%28pi%2C%28pi%29%29%29 Break up the exponent using the identity b%5E%28x%2By%29=b%5Ex%2Ab%5Ey


pi%5E%28log%28pi%2C%282%29%29%29%2Api%5E1 Evaluate log%28pi%2C%28pi%29%29 to get 1

Since we cannot simplify the expression log%28pi%2C%282%29%29 any further, we cannot simplify the entire expression any further.