*The base unit is the number being raised to a power. When you see ln, it means natural logarithm (we'll define natural logarithms below).*In this course only base ten and natural logarithms will be used.

*The base unit is the number being raised to a power. When you see ln, it means natural logarithm (we'll define natural logarithms below).*In this course only base ten and natural logarithms will be used.

We don't really "cancel out" the logs, but that is the effect (only if they have the same base). One of the best ways to understand this problem is to see what's going on using some graphs.

Recall we can only find the logarithm of positive numbers.

The population at the beginning of 2000 was just over `6` billion.

After how many more years will the population double to `12` billion?

We're asked to solve the log of x plus log of 3 is equal to 2 log of 4 minus log of 2. So we have the log of x plus the log of 3 is equal to 2 times the log of 4 minus the log of 2, or the logarithm of 2. Whenever you see a logarithm written without a base, the implicit base is 10.

So we could write 10 here, 10 here, 10 here, and 10 here.

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.

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.

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