Question

Mike was working on solving the exponential equation 37^x = 12; however, he is not quite sure where to start. Using complete sentences, describe to Mike how to solve this equation and how solving would be different if the bases were equal.

Answer 1

12 does not have a base, but to solve
\[37^x=12\] in one step you can write one of two things
1) \(x=\log_{37}(12)\) which actually tells you nothing, just restates the problem or
2) \(x=\frac{\log(12)}{\log(37)}\) by the change of base formula

Answer 2

how do you do #2?

Answer 3

not sure what you mean "how do you do"
if you want a decimal for the second one you use a calculator, or wolfram hold on i will link

Answer 4

http://www.wolframalpha.com/input/?i=log%2812%29%2Flog%2837%29
looks like about \(.688\)

Answer 5

I mean can you show me how do the second choice..

Answer 6

Can you show me how to do both equations ?

Answer 7

you mean how i got from
\[37^x=12\] to
\[x=\frac{\log(12)}{\log(37)}\] or how i computed that number?

Answer 8

actually let me answer both questions
first is it always the case that if
\[b^x=A\] then \[x=\frac{\log(A)}{\log(b)}\] it is the log of the total divided by the log of the base
that is how i went from
\[37^x=12\] directly to
\[x=\frac{\log(12)}{\log(37)}\]

Answer 9

both

Answer 10

the question of how you get a number out of this is by using a calculator

Answer 11

for the first one turning
\[37^x=12\] in to
\[\log_{37}(12)=x\] is because
\[b^x=A\iff \log_b(A)=x\]

Answer 12

you should be able so switch between those forms easily
for example
\[2^x=8\iff \log_2(8)=3\] and
\[10^{-3}=.001\iff\log_{10}(.001)=-3\]

Answer 13

I am so confused... I have to put this in complete sentences and I don't understand what you're doing therefore I can't put it in complete sentences..

Answer 14

lets go real slow

Answer 15

the first thing you have is \(37^x=12\) right ?

Answer 16

right

Answer 17

ok
now you have no idea what number to raise 37 to to get 12,

Answer 18

we can simply rewrite the statement
\[37^x=12\] as an equivalent statement
\[\log_{37}(12)=x\] they say exactly the same thing
and you don't know either of them

Answer 19

before we go on, lets do one we do know
we know
\[2^4=16\] right?

Answer 20

yea

Answer 21

that means
\[\log_2(16)=4\] they say exactly the same thing, they just look different

Answer 22

also we know
\[10^{-3}=\frac{1}{10^3}=0.001\] that means
\[\log_{10}(.001)=-3\] again they say the same thing, they just look different

Answer 23

unfortunately
\[x=\log_{37}(12)\] which is the same as
\[37^x=12\] is not a number you know, so you will not be able to compute this number without a calculator
on your calculator you have log base ten not log base 37

Answer 24

ohk.. I think I understand.

Answer 25

so what are you going to do to come up with \(\log_{37}(12)\)? you have to use the "change of base formula" and put
\[x=\frac{\log(12)}{\log(37)}\]which you can compute using a calculator

Answer 26

.6881