# 0.9 Logarithms  (Page 4/4)

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Interest calculations can be quantified by mathematical formulas. Suppose that we are concerned with making an investment of P dollars at an annual rate of interest of i for n years. Here P denotes the present value of the investment, n represents the number of years the money is to be invested, and i is the interest rate per annum. Let us denote by F the future value of the investment at the end of n years. The value of F can be calculated via the formula

$F=P\left(1+i{\right)}^{n}$

Let us illustrate the use of this formula by means of a problem.

Question: A couple has just had a baby. The couple wishes to make an investment in an account that will be used to fund college expenses. The couple visits an investment advisor and establishes a tax-free college savings account. By tax-free, we mean an account where income tax is not paid yearly. The couple is advised that this feature will enable the account to grow faster.

The couple deposits $25,000 in the account. The couple is told that the value of the account will appreciate at an annual rate of 8% . What will be the value of the account when the child turns 18 years of age? Solution: We are asked to find the value of F . From the problem statement, P is$25,000, i is 0.08 and n is 18. We substitute into the interest formula to obtain the result

$F=\left(\text{25},\text{000}\right)\left(1+0\text{.}\text{08}{\right)}^{\text{18}}=\left(\text{25},\text{000}\right)\left(1\text{.}\text{08}{\right)}^{\text{18}}=\left(\text{25},\text{000}\right)\left(3\text{.}\text{996}\right)=\text{99},\text{900}$

We conclude that the account will be worth $99,900 at the end of 18 years. Let us now a related problem that involves the use of logarithms. Question: An individual inherits$10,000 from a relative. The individual wishes to invest the sum in a tax-free account. He/she plans to use the proceeds of this account as a down payment on a future purchase of a home. The annual interest rate of the account is 6%,

The individual anticipates that he/she will need at least $20,000 for the down payment. How long will it take for the value of the account to grow to$20,000?

Solution: Once again, we begin with the identification of the parameters of the problem. Here, P is 10,000, i is 0.06, and F is 20,000. We incorporate these values into the interest formula

$\text{20},\text{000}=\left(\text{10},\text{000}\right)\left(1+0\text{.}\text{06}{\right)}^{n}$

Let us divide each side by (10,000) and re-arrange terms

$\left(1\text{.}\text{06}{\right)}^{n}=2$

Now let us take the logarithm of each side of the equation. We will use 1.06 as the base of the logarithm

$n={\text{log}}_{1\text{.}\text{06}}\left(2\right)$

The base 1.06 logarithm is related to the base 10 logarithm as follows

${\text{log}}_{1\text{.}\text{06}}\left(x\right)=\frac{{\text{log}}_{\text{10}}\left(x\right)}{{\text{log}}_{\text{10}}\left(1\text{.}\text{06}\right)}$

We will use this relationship to help us find the value for n

$n=\frac{{\text{log}}_{\text{10}}\left(2\right)}{{\text{log}}_{\text{10}}\left(1\text{.}\text{06}\right)}=\frac{0\text{.}\text{301}}{0\text{.}\text{0253}}=\text{11}\text{.}\text{90}$

So the individual should plan on waiting 11.90 years for the account to grow to a value of \$20,000.

Problems such as the previous one often make use of the relationship

${\text{log}}_{a}\left(x\right)=\frac{{\text{log}}_{b}\left(x\right)}{{\text{log}}_{b}\left(a\right)}$

This relationship allows one to convert from the logarithm of any base to a logarithm of another base. Typically, scientific calculators are only able to compute base 10 and natural (base e ) logarithms. One should become acquainted with the conversion of logarithms of other bases to base 10 or base e logarithms.

## Exercises

1. The power of the signal entering an amplifier is 15 mW. The power of the signal that leaves the amplifier is 25 W. Express the gain of the amplifier in decibels.
2. Unlike an amplifier, some devices reduce the power of input signals. This process is called attenuation. Suppose that the power that enters an attenuator is 60 W and the power at the output is 0.9 W. Express the gain of the attenuator in decibels.
3. Consider a signal processing scheme such as that shown in Figure 2. The power of the input signal before it passes through the noisy channel is 40 W. After passing through the noisy channel, the original signal is corrupted by noise. The noise component has a power of 10 W. What is the Signal to Noise ratio of the signal that emerges from the noisy channel?
4. Consider the situation described in exercise 3. The noisy signal enters a signal processor. The signal processor diminishes the noise power of the signal by 5%, while diminishing the power in the noise component by 95%, What is the SNR of the output of the signal processor?
5. Consider the circuit shown in Figure 3. Let us replace the resistor with another whose value is 200 kΩ. (a) What is the new value of the time constant of the circuit. (b) Find the value of the transient response when t = 0.8 seconds. (c) Determine the value of time at which the transient response decays to a value of 1 Volt.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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Can someone give me problems that involes radical expressions like area,volume or motion of pendulum with solution By Inderjeet Brar By Miranda Reising By By Abby Sharp By OpenStax By By Stephen Voron By Ryan Lowe By Brooke Delaney By