Question 280972
{{{2^x = 1/8}}}
The quick solution to this one is to answer the question: "What power of 2 is 1/8?". If you understand exponents and powers of 2 this is a question you can answer.<br>
Otherwise we can solve this using logarithms. To get the simplest exact expression for the solution we will use base 2 logarithms:
{{{log(2, (2^x)) = log(2, (1/8))}}}
On the left side we can use the property of logarithms, {{{log(a, (p^q)) = q*log(a, (p))}}}, to move the exponent out in front. This property and its ability to move the exponent out in front is the very reason we use logarithms on problems like this: to get the variable out of the exponent.
{{{x*log(2, (2)) = log(2, (1/8))}}}
Since {{{log(2, (2)) = 1}}} be definition the left side simplifies to:
{{{x = log(2, (1/8))}}}
Now we just have to simplify the right side. We can use the property of logarithms, {{{log(a, (p/q)) = log(a, (p)) - log(a, (q))}}}, to separate the 1 and 8 of 1/8 into separate logarithms. (You'll see why in a moment.)
{{{x = log(2, (1)) - log(2, (8))}}}
Since any number (except zero) to the zero power is 1, the first logarithm on the right is zero:
{{{x = 0 - log(2, (8))}}}
which simplifies to:
{{{x = -log(2, (8))}}}
I hope now you can figure out the remaining logarithm. What power of 2 results in 8? Answer: 3. So
x = -3<br>
-3, by the way, was the answer to the original, quick-solution question: "What power of 2 is 1/8?" Look at this for a bit to see if this makes sense now.