Question

how do you evaluate 36^(log6 100) without using a calculator?

Answer 1

Hmm this one is a little tricky. You have to apply a bunch of rules of exponents and logs.

Answer 2

We'll first start by rewriting the base as a square. We want to do this so the base of the exponent matches the base of the log, we'll be able to do something with that later.
\[\huge (36)^{(\log_6{100})} \qquad \rightarrow \qquad (6^2)^{(\log_6{100})}\]
Ah hah! They're both 6 now.

Answer 3

\[\huge (a^b)^c \quad = \quad a^{bc}\]
Applying this rule of exponents give us,\[\huge (6^2)^{(\log_6{100})} \qquad \rightarrow \qquad \left(6^{2(\log_6{100})}\right)\]

Answer 4

We can then move the 2 inside of that log applying another rule of exponents.\[\huge a\cdot \log(b) \quad = \quad \log(b^a)\]Applying this to our problem gives us,\[\huge 6^{2(\log_6{100})} \qquad \rightarrow \qquad 6^{(\log_6{100^2})}\]

Answer 5

The next part is a little tricky. You need to recall that exponentiation and the logarithm are inverse operations of one another. So they'll basically cancel out.
Example:\[\huge a^{\log_a(b)}=b\]Applying this to our problem gives us,\[\huge 6^{(\log_6{100^2})} \qquad \rightarrow \qquad 100^2\]

Answer 6

Understand any of that blue? :o

Answer 7

Yes thank you :) i was so stumped with this on my quiz