Logarithms

اللوغارتمات

2 - 1 Definition of Logarithms تعريف اللوغارتمات

• If x = b y then y = logbx

where x and b are positive numbers and b ≠ 1

• logbx is read as “logarithm of x to the base of b”.

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2 - 2 Common and Natural Logarithms اللوغارتم اإلعتيادي و اللوغارتم الطبيعي

• When 10 is the base of a logarithm then it’s called a common logarithmand the base 10 is usually not written.

• When e is the base of a logarithm then it’s called a natural logarithmand it is usually denoted by the letters ln.

Common logarithm of x = log10

x = logx

Natural logarithm of x = log e x = ln x

• Where e is a mathematical constant = 2.718281828459045235…

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2 - 3 Laws of Logarithms قوانين اللوغارتمات

• The fundamental laws of logarithms are:

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Example 1: Change the following from exponential form to logarithmic form:

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Solution:

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Example 2: Change the following from logarithmic form to exponential form:

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Solution:

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Example 3: Evaluate the following expressions, rounded to 3 decimal places:

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Solution:

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Example 4: Solve the following logarithmic equations:

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Solution:

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Example 5: Solve the following logarithmic equations:

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Solution:

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Example 6: Solve the following exponential equations:

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Solution:

By taking the log10 for both side: Taking the log10 for both side gives:

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Example 7: Express each of the following expressions as a single logarithm:

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Solution:

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2 - 4 Population Growth نمو المجتمعات

• The model of many kinds of population growth, whether it be apopulation of people, bacteria, cellular phones, or money isrepresented by the following function:

P(t) = P0 e k t

Where:

P0 = the population at time 0,

P(t) = the population after time t,

k = exponential growth rate, and k > 0

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Example 8: In 2002, the population of India was about 1034 million and the

exponential growth rate was 1.4% per year.

(a) Find the exponential growth function.

(b) Estimate the population in 2008.

(c) After how long will the population be double what it was in 2002?

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Solution:

(a) At t = 0 (2002), the population was 1034 million, then P0 =1034.

k = 1.4%, or 0.014

Therefore, the exponential growth function for this population is:

P(t) = 1304e 0.014 t

Where t is the number of years after 2002 and P(t) is in millions.

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(b) In 2008, t = 6

(c) We are looking for the time T for which P(T) = 2 1034 = 2068.

To find T, we solve the equation:

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2 - 5 Interest Compounded Continuously الفائدة المركبة المستمرة

• If an amount P0 is invested in a savings account at interest rate kcompounded continuously, then the amount P(t) in the accountafter t years is given by the exponential function:

P(t) = P0 e k t

where:

P0 = the amount at time 0,

P(t) = the amount after time t,

k = continuously compounded interest rate.

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Example 9: Suppose that $2000 is invested at interest rate k, compounded continuously, and grows to $2504.65 in 5 years.(a) What is the interest rate?

(b) Find the exponential growth function.

(c) What will the balance be after 10 years?(d) After how long will the $2000 be doubled?

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Solution:

(a) P(0) = P0 = $2000

Thus, the exponential growth function is:

P(t) = 2000e kt

Since P(5) = $2504.65,

Then 2504.65 = 2000 e k(5)

2504.65 = 2000 e 5k

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(b) Substituting 0.045 for k in the function P(t) = 2000e kt gives:

P(t) = 2000e 0.045t

So, the interest rate is 0.045 or 4.5%.

(c) The balance after 10 years is:

P(10) = 2000e 0.045(10)

= 2000e 0.45

= $3136.62

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So, the original investment of $2000 will double in about 15.4 years

(d) To find the time T needed for doubling the $2000,

we set P(T) = 2 × P0 = 2 × $2000 = $4000 and solve for T.

4000 = 2000e 0.045T

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