We cannot simplify any further. The final answer is:
Simplify, without use of a calculator:
8 can be written as
. 32 can be written as
.
We can use:
We can now use
The final answer is:
Write
as the logarithm of a single number.
)
Exponent rule:
Solving simple log equations
In grade 10 you solved some exponential equations by trial and error, because you did not know the great power of logarithms yet. Now it is much easier to solve these equations by using logarithms.
For example to solve
in
correct to two decimal places you simply apply the following reasoning. If the LHS = RHS then the logarithm of the LHS must be equal to the logarithm of the RHS. By applying Law 5, you will be able to use your calculator to solve for
.
Solve for
:
correct to two decimal places.
In general, the exponential equation should be simplified as much as possible. Then the aim is to make the unknown quantity (i.e.
) the subject of the equation.
For example, the equation
is solved by moving all terms with the unknown to one side of the equation and taking all constants to the other side of the equation
Then, take the logarithm of each side.
Substituting into the original equation, yields
Similarly,
is solved as follows:
Substituting into the original equation, yields
Solve for
in
There are two possible bases: 5 and 7.
is an exponent of 5.
In order to eliminate 7, divide both sides of the equation by 7 to give:
Exercises
Solve for
:
Logarithmic applications in the real world
Logarithms are part of a number of formulae used in the Physical Sciences. There are formulae that deal with earthquakes, with sound, and pH-levels to mention a few. To work out time periods is growth or decay, logs are used to solve the particular equation.
A city grows 5% every 2 years. How long will it take for the city to triple its size?
Assume
, then
.
For this example
represents a period of 2 years, therefore the
is halved for this purpose.
So it will take approximately 45 years for the population to triple in size.
I have R12 000 to invest. I need the money to grow to at least R30 000. If it is invested at a compound interest rate of 13% per annum, for how long (in full years) does my investment need to grow ?
In this case we round up, because 7 years will not yet deliver the required R 30 000.
The investment need to stay in the bank for at least 8 years.
Exercises
The population of a certain bacteria is expected to grow exponentially at a rate of 15 % every hour. If the initial population is 5 000, how long will it take for the population to reach 100 000 ?
Plus Bank is offering a savings account with an interest rate if 10 % per annum compounded monthly. You can afford to save R 300 per month. How long will it take you to save R 20 000 ? (Give your answer in years and months)
End of chapter exercises
Show that
Show that
Without using a calculator show that:
Given that
and
Write
in terms of
Express
in terms of
Express
in terms of
and
Simplify, without the use of a calculator:
Simplify to a single number, without use of a calculator:
Given:
and
Express
in terms of
.
Hence, or otherwise, find
in terms of
and
.
Given:
Prove:
Evaluate without using a calculator:
If
, determine,
without using a calculator :
Given:
Determine the values of
for which
is defined.
Solve for
if
.
Solve:
(Answer(s) may be left in surd form, if necessary.)
Find the value of
without the use of a calculator.
Simplify By using a calculator:
Write
in terms of
and
if
and
.
Calculate:
Solve the following equation for
without the use of a calculator and using the fact that
Solve the following equation for
:
(Give answer correct to 2 decimal places.)
Questions & Answers
Discuss the differences between taste and flavor, including how other sensory inputs contribute to our perception of flavor.
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include:
1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body
the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature