Final answer:
The domain of an exponential function is all real numbers, while the domain of a logarithmic function is limited to positive real numbers. In solving logarithmic equations, common logarithm and natural logarithm rules such as the product rule, quotient rule, and power rule are essential.
Step-by-step explanation:
Finding the domain of exponential and logarithmic functions is crucial as it defines the set of all possible input values (x-values) for which the function is defined. For exponential functions, the domain is all real numbers because you can raise a number to any power. However, for logarithmic functions, the domain is more restrictive; only positive numbers can be used as input because you cannot take the logarithm of a negative number or zero.
Regarding the common logarithm, remember that \(\log(x)\) gives us the power to which 10 must be raised to get x. Hence, the exponential function \(10^y = x\) and its logarithmic form \(\log(x) = y\) are inverses of each other. For natural logarithms (ln), they are concerned with the number \(e\) rather than 10, but still, they only accept positive numbers as inputs; hence ln(x) is only defined for x>0.
For example, \(\log(100)\) is 2 because \(10^2 = 100\). Using properties of logarithms such as the product rule (\(\log(xy) = \log(x) + \log(y)\)), the quotient rule (\(\log(x/y) = \log(x) - \log(y)\)), and the power rule (\(\log(x^a) = a\log(x)\)), we can simplify and solve complex logarithmic equations.