Exponential equations
Sometimes a variable may be in a power. In such cases, the equation is called exponential equation. We need logarithms to take the unknown variable down from the power.
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From the definition, we can easily see some basic properties of logarithmic:
Logarithms are sometimes thought to be difficult concept. In fact, they are very easy, if you read them right. For example, is read: to which power should the a be raised to get x.
Commonly used symbol for logarithmic of base e is ln and for logarithmic of base 10 is lg . These symbols are used during the lessons. Check your own calculator, how they are denoted there.
Example 1.
Example 2.
There are only three basic formulas for handling logarithms and one transformation formula for changing the base.
If x > 0, y > 0 and r is a real number, then
You can read the formulas also from the right to the left.
The base of a logarithm can be changed with the formula
where a is the original base and b is the new base.
Example 3.
Let us solve
Because both sides are positive , we can take the natural logarithm. By
doing it, we get the variable down from the exponent and it can be
solved.
Example 4.
Let us solve
Because both sides are positive , we can take the natural logarithm. By
doing it, we get the variable down from the exponent and it can be
solved.
Let us solve
Because both sides are positive , we can take the natural logarithm. By
doing it, we get the variable down from the exponent and it can be
solved.