How can an exponential equation be changed to a logarithmic equation and why is it done?
An exponential equation is one of the form y = a*b^x, where a is the initial value of y when x = 0 and it increases at a non-linear rate. The value of y increases by the factor b, also called the base, for every unit increase in x.
It is often required to determine what the value of x should be for y to take on a particular value or if the value of y is known, what the corresponding value of x is. Examples of this could include the time required for a certain radioactive substance to decay till a particular fraction remains or how long will it take the population of an organism that grows at an exponential rate to reach a certain number.
To solve for values of x, the use of logarithms makes the process very easy. The properties of logarithms that helps in this are log (a^b) = b*log a and log (a*b) = log a + log b.
For an exponential equation y = a*b^x, we take the log of both the sides
=> log y = log (a*b^x)
=> log y = log a + log b^x
=> log y = log a + x*log b
=> x = (log y - log a)/log b
(If the base b is "e," ln is usually used.)
Now, we have the value of x in the form of logarithms the values of which are available in standard tables.
Exponential equations are converted to logarithmic equations to determine the variable x in an easy manner.