# What are common mistakes students make with logarithms?

Sep 2, 2014

Students make mistakes with logarithms because they are working with exponents in reverse! This is challenging for our brains, since we are often not so confident with our powers of numbers and the exponent properties...

Now, powers of 10 are "easy" for us, right? Just count the number of zeros to the right of the "1" for positive exponents, and move the decimal to the left for negative exponents....

Therefore, a student who knows powers of 10 should be able to do logarithms in base 10 just as well:
log(10) = 1 which is the same as ${\log}_{10} \left(10\right) =$ 1
log(100) = 2
log(1000) = 3
log(10000) = 4
log(1) = 0
and so on. Did you notice that we mathematicians are so lazy that we don't even bother to show the BASE 10? On top of that, we assume that everyone knows and understands that key to understanding!

But, let's try some other bases:
${2}^{3} = 8$ so ${\log}_{2} \left(8\right) = 3$ since the answer to the logarithm is the power of 2 that equals 8.

The answer to a log is the exponent....hmmm....

${3}^{4} = 81$ so ${\log}_{3} \left(81\right) = 4$
3 to the fourth power is 81, so the log in base 3 of 81 equals 4.
Remember, BASE 3. And the answer is the power!!

Last one: ${4}^{-} 1 = \frac{1}{4}$ so ${\log}_{4} \left(\frac{1}{4}\right) = - 1$
Keep working!!