g 1 x 2 =log 3 xdefines the logarithmic function with base 3.10.4 Properties of Logarithms
Product Rule
Quotient Rule
Power Rule
Special Properties
blog^ b^ xx and logb bxxloga xrr loga xlogax
yloga xloga yloga xyloga xloga yCONCEPTS EXAMPLES
Product ruleQuotient rulePower rule6 log^610 = 10 log 3 34 = 4 Special propertieslog 10 23 = 3 log 10 2log 59
4
=log 5 9 - log 5 4log 2 3 m=log 2 3 +log 2 m10.5 Common and Natural Logarithms
Common logarithms (base 10)are used in applications
such as pH, sound level, and intensity of an earthquake.
Natural logarithms (base e)are often found in formulas
for applications of growth and decay, such as time
for money invested to double, decay of chemical
compounds, and biological growth.
Change-of-Base Rule
If then
loga xlogb x
logb a.
a 7 0, aZ1, b 7 0, bZ1, x 7 0,10.6 Exponential and Logarithmic
Equations; Further Applications
To solve exponential equations, use these properties
1.If then bx=by, x=y.
1 b 7 0,bZ 12.
Use the formula to find the pH (to one decimal
place) of grapes with hydronium ion concentration
Substitute.
Property of logarithms
Evaluate with a calculator.Use the formula for doubling time (in years) to find
the doubling time to the nearest tenth at an interest rate of 4%.Substitute.Evaluate with a calculator.
The doubling time is about 17.7 yr.log 3 17 =ln 17
ln 3=
log 17
log 3L2.5789
L17.7
t 1 0.04 2 =ln 2
ln 11 +0.04 2t 1 r 2 =ln 11 ln 2+r 2L4.3
=- 1 log 5.0+log 10-^52pH=-log 1 5.0* 10 -^525.0* 10 -^5.
pH=-log 3 H 3 O+ 4For defines the
logarithmic function with base a.
Graph of
1.The graph contains the point
2.When the graph rises from left to right. When
the graph falls from left to right.
3.The y-axis is an asymptote.
4.The domain is 1 0, q 2 , and the range is 1 - q, q 2.
06 a 6 1,a 7 1,1 1, 0 2.
g 1 x 2 loga xa 7 0, aZ1, x 7 0, g 1 x 2 loga x(^013)
x
–2
1
y
g(x) = log 3 x
Solve.
Set exponents equal.
Divide by 3.
The solution set is E^53 F.
x=
5
3
3 x= 523 x= 25(continued)