## 10.2 Outline

- Fundamental properties
- logarithmic equation
- Grant’s tomb properties
*x*is the exponent on a base*b*that gives bx.- log
is the exponent on a base_{b}x*b*that gives*x*.

- Logarithmic equations
- log of both sides theorem
- four types
- the unknown is the logarithm
- the unknown is the base
- the logarithm of an unknown is equal to a number
- the logarithm of an unknown is equal to the logarithm of a number

- Laws of logarithms
- additive law
- subtractive law
- multiplicative law

## 10.2 Essential Ideas

**Fundamental properties of logarithms **(Grant’s tomb properties):

log_{b}b^{x}=x

b^{xlog}*b*^{x }*= x*

**logarithmic equation **is an equation for which there is a logarithm on one or both sides. The key to solving logarithmic equations is the **log of both sides theorem**:

If *A, B, *and *b *are positive real numbers (with *b *not equal to 1), then log * _{b}A = *log

_{b}*B*is equivalent to

*A = B.*

**Laws of Logarithms**

If *A, B, *and *b* are positive numbers, *p* is any real numbers and *b* is not equal to 1:

**Addition Law**:

log_{b}(*AB*) = log* _{b}A + *log

_{b}B

**Subtraction Law**:

log_{b}(*A/B*) = log* _{b}A – *log

_{b}B

**Multiplication Law**:

log_{b}*A ^{p}* =

*p*log

_{b}A