Find the exponential value of a number. I'm wondering if there is a way to add two numbers with the same base but different exponents without converting the base of the numbers. For example lets say your adding 2^2 and 2^4 the value of 2^2 is 4 and the value of 2^4 is 16 so the total of the two numbers is 20. The closest I could get my calculator to finding a base of 2 with a value of 20 was 2^4.3219281 which equaled 20.000000071. So I have a few questions. 


*

*Can 2 be raised to a power that's exactly equal to 20?

*I know that with division and multiplication you can just add or subtract the exponents. Are there any similar tricks for adding or subtracting values with exponents?

*Is there anyway to arrive at the result, or a similar result of 2^4.3219281 without changing the base in the process?
 A: *

*There is a particular function just designed for solving problems of the form $b^x=a$.  The solution to said problem is given by$$x=\log_b(a)=\frac{\log(a)}{\log(b)}$$where we use the second form if your calculator doesn't specify the base of the logarithm.  For example,$$2^x=20\implies x=\log_2(20)$$

*No, definitely no simple tricks.  $\ddot\frown$

*As to calculating what $\log_2(20)$ is without a calculator, one can implement the following algorithm:$$2^x=20=5\times2^2=2^{\log_2(5)}2^2=2^{2+\log_2(5)}$$$$x=2+\log_2(5)$$So the problem now is to find $\log_2(5)$, which is much easier to approximate.  Clearly, it's larger than $2$, so maybe $\log_2(5)\approx2.3$, giving us $x=4.3$
A: You seem to be looking for an exponent $f(a,b)$ with the property that
$$2^{f(a,b)} =2^a+2^b$$
The simple answer is that you want to take 
$$f(a,b) = \log_2(2^a+2^b)$$
But that's just the definition of the logarithm ($x^y=z\iff y=\log_x z$).
Other than this, there's not any real way to do the conversion.
In this specific case, you used the number $\log_2 20\approx 4.32193$.
A: There is no "trick" for $\log_2 (2^a + 2^b)$, other than
$$ \log_c (c^a + c^b) = \log_c (c^a (1 + c^{b-a})) = a + \log_c(1 + c^{b-a})$$
In particular $\log_2(20) = 2 + \log_2(5)$.  
$\log_2(5)$ is a transcendental number by the Gelfond-Schneider theorem.
