Question 711389: Evaluate the given expressions (to two decimals places.)
(a) log 19.1
(b) log2 goes at the bottom of g.and the number 8
(c) log5 1
(d) 0.042
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! Some logs can be evaluated "by hand" (without a calculator). For these logs it is better to evaluate them by hand because:- The answer will be exact when you do it by hand. When you use a calculator you often get either a decimal approximation or the decimal version of some simple fraction.
- Doing logs by hand exercises and strengthens your understanding of what logs are.
- Your calculators usually know how to do 1 or 2 types of logarithms:
- Base 10 logs, "log"
- Base e (or natural) logarithms, "ln".
To use a calculator to find logs of any other base, like parts "b" and "c" of your problem, you will have to convert the logs into base 10 or base e logs using the base conversion formula:
So we are going to try to do each log by hand first before we reach for our calculator. If we are unable to find the log by hand, we will resort to a calculator.
a) log(19.1)
To do logs by hand you must understand what they are. In general, logarithms are exponents. This particular log is a base 10 log. Base 10 logs, in general, are exponents that one would put on a 10. This specific base 10 log, log(19.1), represents the exponent one would put on a 10 to get a result of 19.1. So to be able to do this log by hand we would have to be able to figure out what power of 10 is equal to 19.1. Since every power of 10 that I know of only has digits of 1's and 0's, I have no idea what power of 10 would result in 19.1. So this is a case for our calculators. Since most calculators have a button for base 10 logs, "log", this should be easy. I get:
log(19.1) = 1.2810333672477275376350435982706
Rounded to two decimal places this would be:
log(19.1) = 1.28
b) 
(Note: This is called "the base 2 log of 8" (which is a much better way to describe it than the way you posted this.) This base 2 log is an exponent one would put on a 2 to get a result of 8. If you don't know what power of 2 result in 8, don't immediately reach for your calculator. Explore some powers of 2 first. It should not take long to find that  . So:
c) 
This base 5 log represents an exponent one would put on a 5. Specifically, this base 5 log is the exponent one would put on a 5 to get a result of 1. This is another one we can do by hand. We should know that any number (except 0) raised to the zero power is equal to 1. So the log of 1, no matter what the base, is always zero:
d) 0.042
???. If this is correct then all we have to do is round to two decimal places:
0.04
P.S. If you are using the calculator program on Windows but you do not see a button for "log" or "ln":- Start the calculator program. It is often found under "accessories".
- Click on the "View" menu.
- Click on "Scientific" in the list that appears.
This should change the display so that it has many more buttons, including "log" and "ln".
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