Section 9.6

• There are two logarithmic bases that occur so frequently in applications that they are given special names.

• Common logarithms are logarithms to base 10.

• Natural logarithms are logarithms to base e (an irrational number which is approximately equal to 2.7183).

Common Logarithms

log x means log10 x

You can use a calculator to approximate common logarithms using the button below.

LOG

To find exact values of common logarithms, use the definition of logarithms to rewrite the expressions in exponential form to evaluate.

Example

Find the exact value of each of the following logarithms.

1) log 10,000

log 104 = 4

2) log 0.001

log 10-3 = -3

3) log 10

log 10½ = ½

Example

Solve the following equation for the variable. Give both an exact answer and an answer approximated to four decimal places.

log 3 1.3x 1.310 3x

1.310

3x (exact answer)

6.6509x (approximate answer)

One of the most popular uses of common logarithms involves the Richter scale for measuring the intensity of earthquakes.

For R (magnitude of the earthquake), a (amplitude in micrometers of the vertical motion of the ground at the recording station), T (number of seconds between successive seismic waves), and B (adjustment factor that takes into account the weakening of the seismic wave as the distance increases from the epicenter of the earthquake), the formula is . . .

loga

R BT

Example

Find the intensity R of an earthquake when the amplitude a is 300 micrometers, time T between waves is 2.5 seconds, and B is 2.6. Round the answer to one decimal place.

loga

R BT

300log 2.6

2.5R

log 120 2.6R 4.7R

Natural Logarithms

ln x means loge x

You can use a calculator to approximate natural logarithms using the button below.

LN

To find exact values of common logarithms, use the definition of logarithms to rewrite the expressions in exponential form to evaluate.

Example

Find the exact value of each of the following logarithms.

1) ln e4

4

32) ln e

⅓

Example

Solve the following equation for the variable. Give both an exact answer and an answer approximated to four decimal places.

ln (2 5) 3.4x 3.4 2 5e x

3.4 5

2

ex

(exact answer)

12.4821x (approximate answer)

• Logarithms have many uses in applications.

• However, most calculators only have the ability to calculate common or natural logs, but not logarithms to other bases.

• Therefore, we need to be able to change the base of our logarithms so that we can approximate them, when necessary.

Change of Base

If a, b, and c are positive real numbers and neither b nor c is 1, then

log

loglog

cb

c

aa

b

Example

Approximate log5 to four decimal places.1

6

5

1log 1 6log

6 log 5

5

1log 1.1133

6