Rewriting and solving a real-world exponential model
The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation
represents this situation, where
is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?
We begin by rewriting the exponential equation in logarithmic form.
Next we evaluate the logarithm using a calculator:
The amount of energy released from one earthquake was
times greater than the amount of energy released from another. The equation
represents this situation, where
is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?
The most frequently used base for logarithms is
Base
logarithms are important in calculus and some scientific applications; they are called
natural logarithms . The base
logarithm,
has its own notation,
Most values of
can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base,
For other natural logarithms, we can use the
key that can be found on most scientific calculators. We can also find the natural logarithm of any power of
using the inverse property of logarithms.
Definition of the natural logarithm
A
natural logarithm is a logarithm with base
We write
simply as
The natural logarithm of a positive number
satisfies the following definition.
For
We read
as, “the logarithm with base
of
” or “the natural logarithm of
”
The logarithm
is the exponent to which
must be raised to get
Since the functions
and
are inverse functions,
for all
and
for
Given a natural logarithm with the form
evaluate it using a calculator.
Press
[LN] .
Enter the value given for
followed by
[ ) ] .
Press
[ENTER] .
Questions & Answers
if three forces F1.f2 .f3 act at a point on a Cartesian plane in the daigram .....so if the question says write down the x and y components ..... I really don't understand
a fixed gas of a mass is held at standard pressure temperature of 15 degrees Celsius .Calculate the temperature of the gas in Celsius if the pressure is changed to 2×10 to the power 4
