Word Problems of Logarithmic Function

Introduction to Word problems of logarithmic function:

A logarithm is simply an exponent. We can write this logarithm form as follows:

Log53= 2. We say this logarithm of 3 to the base 5 is 2.

We defined the logarithmic function as:

f (x) = log b x

Here the b is base of the logarithms. The logarithmic functions have more real life applications like that acoustics, electronics, and earthquake analysis and population prediction etc...

To Solve Word Problems of Logarithmic Function:

1) The invested amount is $1500 in the account and the annual rates compounded quarterly, and you wanted to have $3,000 in the account at the end of the investment time, what interest rate time were 1 year?

Solution:

$3000 = $1500`(1+r/4)^(4t)`

Here $3000 is the balance

The starting investment=$1500

Let t= the number of years

Let annual percentage rate=r

The annual rate of r% is converted to a quarterly interest rate. The compounding is quarterly . The exponent is 4t because there are 4 compounding periods in every year.

$3000 = $1500(1+`r/4` )4t

To derive r:

$3000 = $1500`(1+r/4)^(4t)`

Divide by$1,500 both sides

2= `(1+r/4)^(4xx1)`

Take both sides by natural logarithm

Ln (2)=Ln(1+r/4)4.1

Ln (2) =4Ln `(1+r/4)`

Divide  both sides by 4

0.693147181/4=ln `(1+r/4)`

0.173286795= ln `(1+r/4)`

e 0.173286795= `(1+r/4)`

1.18920711=`(1+r/4)`

Subtract both sides by 1

0.18920711=`r/4`

Multiply both sides by 4

0.75682844= r

To solve Word problems of logarithmic function

Earthquake in San Francisco registered 9.5 using Richter scale. In the same year, another earthquake was recorded in North America that was four time stronger. What was the magnitude of the earthquake in North American?

Solution:

Convert the sentence to an equivalent mathematical sentence MNF= log (ISF/S) =9.5

9.5 = = log `(I^(SF)/S)`

MNA=  log `(I^(SA)/S)`

ISA=4ISF

MNA= log `((4I^(SF))/S)`

Solve for MNA.

MNA   =log `((4I^(SF))/S)`

=log 4 ISF-log S

= log4+( log ISF – log S)

= log 4 +log `((I^(SF))/S)`

= log 4 +9.5     

= 0.602059991+9.5

= 10.10206

MNA =10.10206

Example Word Problemswith Solutions:

To solve Word problems of logarithmic function

A recent earthquake measured 9.2 using Richter scale. How many times more intense was this earthquake than an earthquake that measured 4.3 using Richter scale?

Solution:

Let A refer to the earthquake that measured 9.2

B refer to the earthquake that measured 5.5

9.2 = log `A/S`

5.5 = log `B/S`

Solve for `A/B`

9.2-5.5 =log `A/S` -log `B/S`

3.7= (log A- log S)-(logB- log S)

3.7= log A- log S-log B+ log S

3.7 = log A- log B

3.7 = log  `A/B`

10 3.7 =  `A/B`

5 011.87234= = `A/B`

`A/B` = 5 011.87234

A= 5 011.87234 B

To solve Word problems of logarithmic function

A telescope is limited in its usefulness by the brightness of the star that it is aimed at and by the diameter of its lens. One measure of a star's brightness is its magnitude; the dimmer the star, the larger its magnitude. A formula for the limiting magnitude L of a telescope, that is, the magnitude of the dimmest star that it can be used to view, is given by

L = 9 + 5.1 log d

where d is the diameter (in inches) of the lens.

A) What is the limiting magnitude of a 3.5- inch telescope?

B) What diameter is required to view a star of magnitude 14?

Given:
L = 9 + 5.1 log(d)

A Solution:

d = 3.5
L = 9 + 5.1 log (3.5)
L ≈ 9 + 5.1(0.5441)
L ≈ 11.7747

B Solution:

L = 14:
14 = 9 + 5.1 log (d)
`5/(5.1)`  = log(d)
`1/(1.02)`  = log(d)
`10^(1/1.02)` = d
9.5586 ≈ d