Relating Logarithms to Exponential Equations
There is an inverse relationship between logarithms and exponents. For example, if we have the expression 3^{x} = 9, we can gather that x = 2, because 3 squared equals 9. As a logarithmic expression, we can write this equivalently as log_{3}(9) = 2. This reads, "the log, base 3, of 9 is 2." The expression tells us that the base number, 3, must be raised to the power of 2 in order to equal 9.
In general, we can write the relationship between logarithms and exponents as follows:
Notice that x and y switched as being isolated onto one side of the equals sign. This is characteristic of inverse relationships. Also note that the base to the exponential is the base of the logarithm.
Let's use this relationship to re-write the exponential equation 8 = 2^{x} as a logarithmic equation:
If you know your powers of 2, you may be able to gather that x = 3 in this case.
Common Log and Natural Log
If you have a scientific calculator that can compute logarithms, there are likely two kinds of log buttons on your calculator: one that simply says "log" and another that says "ln." The first button, "log" is known as common log, while the other, "ln," is referred to as the natural log. They are both logarithms, but their difference is in their base. Common log operates under a base of 10. So if you ever see expressions such as log(42) or log(67), the base of the log is 10.
Whenever a base is not explicitly written next to "log," it is assumed to be the common log, which is base 10.
The other kind of logarithmic button likely on your calculator is the natural log, or "ln." The abbreviation ln comes from the Latin logarithmus naturali. The base of this logarithm is the mathematical constant e. "e" or Euler's constant is approximately equal to 2.718282. If you have the natural log button on your calculator, definitely use it for the most accurate calculations. Otherwise, use the approximation 2.718282.
ln, or natural log, operates in base e, which is approximately equal to 2.718282. ln(x) and log_{e}(x) are the same expressions.
Evaluate Logarithmic Expressions
We can use the relationship between exponential equations and logarithmic equations to evaluate expressions by thinking about how many times we must multiply a given number by itself to result in another given number.
Evaluate log_{4}(64)
Evaluate log_{3}(243)
The inverse of a power, the logarithm describes how many times a number should be multiplied to result in another number.