SOLUTION: Can you pls help me with the following: Use the Change of Base Formula to evaluate each expression. Leave your answer in "calculator ready" format. log_4⁡8 Thank yo

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Can you pls help me with the following: Use the Change of Base Formula to evaluate each expression. Leave your answer in "calculator ready" format. log_4⁡8 Thank yo      Log On


   

Question 409811: Can you pls help me with the following:
Use the Change of Base Formula to evaluate each expression. Leave your answer in "calculator ready" format.
log_4⁡8
Thank you!!
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%284%2C+%288%29%29
No calculator I know of "knows" base 4 logarithms. So the only way to express this logarithm in a way that your calculator can handle is to use the Change of Base Formula to change this base 4 logarithm into an expression involving logarithms your calculator does "know", like base 10 or base e (aka ln).

The Change of Base Formula is:
log%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28a%29%29
We can use this to change to base 2 logarithm into an expression of base 10 logarithms:
log%28%288%29%29%2Flog%28%284%29%29
or base e logarithms:
ln%2887%29%2Fln%284%29

Note: If you actually calculate both of these expressions you will find, believe it or not, that they are equal!

Since 4 and 8 are both powers of 2, we can also find this logarithm without a calculator at all. Using the Change of Base Formula to write the base 4 logarithm in terms of base 2 logarithms we get:
log%282%2C+%288%29%29%2Flog%282%2C+%284%29%29
Since 8+=+2%5E3 the top logarithm is 3. And since 4+=+2%5E2 the bottom logarithm is 2. So now we have:
3/2 (or 1.5)
Note: log%284%2C+%288%29%29 is exactly 3/2 (or 1.5). If you use your calculators on the base 10 or base e expressions you may not get exactly 1.5, you might get 1.49999999 or 1.5000001 or some other number very, very close to 1.5. This can happen because your calculator uses very close approximations for logarithms.