Final answer:
To write a logarithmic equation in exponential form, convert the base of the logarithm into the base of the exponent, with the logarithm's result as the exponent and its argument as the result. For example, log(100) = 2 becomes 10^2 = 100. This principle is widely applicable in mathematics and sciences.
Step-by-step explanation:
To convert a logarithmic equation to an exponential form, it is essential to understand the properties of logarithms and exponents, as they are inverse functions. The base of the logarithm becomes the base of the exponent in the conversion process. If we have a logarithmic equation of the form logb(x) = y, it translates to by = x in exponential form.
For example, if we take the common logarithm of 100 which is given by log(100) = 2, this means that 10 must be raised to the power of 2 to equal 100. Therefore, in exponential form, it is written as 102 = 100. Another important property to remember is that for any number x, we have eln(x) = x where e is the base of the natural logarithm, and similarly for ln(ex) = x.
Applying these principles, we can express the integrated rate law for a first-order reaction or compute attributes of growth through the use of logarithms and exponentials. When dealing with exponents, remember that the logarithm of a product is the sum of the logarithms (log xy = log x + log y), and the logarithm of a number raised to an exponent is the exponent multiplied by the logarithm of the base number (log(xn) = n · log(x)). These rules are fundamental in mathematics and various scientific disciplines, including chemistry and physics.