Question 382841
I'm guessing your expression is:
{{{3^log(3, (7))}}}
If this is correct then it simplifies to:
7<br>
The reason this is so is based on an understanding of what logarithms are. The idea behind logarithms is that it is possible to take <i>any</i> positive number, raise it to the correct power and get any other positive number as a result. For example, it is possible to raise 7 to some power and get a 12.<br>
Some of these exponents are easy to figure out. For example:
What exponent for 2 results in 4? Answer: 2
What exponent for 3 results in 9? Answer: 2
What exponent for 81 results in 9? Answer: 1/2 (Remember 1/2 as an exponent means square root!)
What exponent for 13 results in 1? Answer: 0 (Remember any non-zero number to the zero power is a 1!)
What exponent for 8 results in 1/8? Answer: -1 (Remember an exponent of -1 means reciprocal!<br>
Logarithms are used to express these exponents. In general {{{log(a, (b))}}} expresses the exponent for a that results in b. So, using the examples above:
{{{log(2, (4)) = 2}}}
{{{log(3, (9)) = 2}}}
{{{log(81, (9)) = 1/2}}}
{{{log(13, (1)) = 0}}}
{{{log(8, (1/8)) = -1}}}<br>
Many of these exponents are hard to know. For example, what power of 14 results in 22? Without knowing exactly what this exponent is we can still express it as:
{{{log(14, (22))}}}<br>
So why is
{{{3^log(3, (7)) = 7}}}?
Well, the exponent, {{{log(3, (7))}}}, stands for <u>the exponent for 3 that results in 7</u>. And where to we find {{{log(3, (7))}}}? Answer: As the exponent on a 3! So if we raise 3 to the power for 3 the results in 7, we will of course get a 7!<br>
In general
{{{a^log(a, (b)) = b}}} 
for <i>any</i> positive numbers a and b!