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Expert: - 7/8/2010


Question
I'm totally lost. I can follow how they do it in the book, but when I try to do one on my own I just don't know where to start. I guess I need clarification, so please help if you can.

Use properties of logarithms to rewrite and simplify the logarithmic expression:

1.log4 8

2. log5 1/250

3. ln(5e^6)

Find the exact value of the logarithmic expression without using a calculator. (If not possible, state reason)

1. log3 9

2. log2 4th root of 8

3. log4 16^2

Answer
The definition of a log is the power needed.
For example, since 3^4 = 81, log3 81 = 4.

1. Note that the squareroot of 4 is 2, and 2 cubed is 8, so the answer is 3/2.

2. Note that log(a/b) = log a - log b, so log5 1/250 = log5 1 - log5 250.
The log 1 is 0 in any base since x^0 = 1, so here we have -log5 250.
Now log ab = logb a + log b, so log5 250 = log5 2 + log5 125.
Since 5*5*5=125, log5 125 = 3, so the answer is -log5 2 - 3.

3. Using ln ab = ln a + ln b, we have ln 5 + ln(e^6)
Knowing the ln(x) and e^x are inverse functions, ln(e^6) = 6,
so the answer is ln 5  +  6, which I write as 6 + ln 5,

1. 9 = 3², so log3 9 = log3 3² = 2

2. 8 is 2 cubed, and exponets can be taken out front on logs, we we have (1/4)3 = 3/4.

3. 4² = 16, and the exponet on value can be taken out front, so we have 2 * log4 4² =
2*2*log4 4 = 2*2*1 = 4.

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