Here's the graph of `y = ln (x 2)^2 - 2`, based on the first expression: So while the Log Law says we can write `ln (x 2)^2` as `2ln (x 2),` they are not really the same function.The population of the earth is growing at approximately `1.3%` per year.
The method you use depends on the type of logarithmic equation you are trying to solve.
Other equations can be simplified using other properties of logarithms.
One difficulty that arises is that eliminating logarithms and solving the resulting equation can introduce spurious solutions.
These solutions violate the principle that the argument of the log function must always be positive.
Population after 3 years: `6\ "billion" × (1.013)^3`.
So our population, P, after t years, is given by: When the world population is 12 billion, the net number of people in the world will be increasing at the rate of about 5 per second, if the growth rate is still 1.3%.[Source] This suggests a growth rate of about 0.6%, much lower than that experienced during the 20th century.A logarithm is the power to which a number must be raised in order to get some other number (see Section 3 of this Math Review for more about exponents).The following graph shows one of the estimates for world population growth during the 21st century.We see that the population will be 11 billion by about 2100!It is generally wise to check solutions by plugging them into the original equation and making sure that both sides are defined. We can think of logarithmic functions as the inverse of exponents. The base of the log is 10, so we must raise both sides of the equation to be powers of 10: On the left hand side, the 10 and log cancel, leaving just 2x.2x = 10,000 x = 5,000 We can check this answer by substituting it back in for x.First we must get the log by itself by moving the 2 and the 5: The base of the log is 3, so we must raise both sides of the equation to be powers of 3: On the left hand side, the 3 and cancel, leaving just x - 1.This lesson will discuss two ways to solve logarithmic equations.