LOGARITHMS Definition of a logarithm If a number y
Description: LOGARITHMS Definition of a logarithm If a number y can be written in the form of ax, the index x is called the logarithm of y to the base a. In other word, if y ax, then logay x where y and a are positive numbers. Example 1: Write the
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slide1. LOGARITHMS<br>
slide2. Definition of a logarithm If a number y can be written in the form of ax, the index x is called the logarithm of y to the base a.
In other word, if y = ax, then logay = x where y and a are positive numbers.
Example 1: Write the exponent 3x = 16 into the logarithm form
Example 2: Write the exponent y-5 = 4.6 into the logarithm form<br>
slide3. Definition of a logarithm (cont) Example 3: Find the value of x in each of the following equations
(a) log3x = 4
(b) log2x = -5
(c) log10(x – 4) = 3<br>
slide4. Properties of Logarithm From the definition of logarithm, we can proceed further to obtain the following properties
(a) loga1 = 0
(b) loga a = 1
(c) loga an = n<br>
slide5. Properties of Logarithm (cont) Example 4: Without using the calculator, evaluate the following.
log101
log2.5(5/2)
log2 25
log381
log40.25
log8 2<br>
slide6. The Rules of Logarithm Proof:
Let loga x = m and loga y = n.
So, x = am and y = an.
Thus, xy = amï‚´ an = a(m+n)
Applying the definition of a logarithm gives<br>
slide7. The Rules of Logarithm (cont) Proof:
Let loga x = m and loga y = n.
So, x = am and y = an.
Thus,
Applying the definition of a logarithm gives<br>
slide8. The Rules of Logarithm (cont) Proof:
Let loga x = m with x = am.
Raising each side to the power r gives
Applying the definition of a logarithm gives<br>
slide9. The Rules of Logarithm (cont) Example 5: Simplify the following as a single logarithm<br>
slide10. The Rules of Logarithm (cont) Example 6: Without using a calculator, evaluate the following.<br>
slide11. The Rules of Logarithm (cont)<br>
slide12. Natural Logarithm The most frequently used bases for log are 10 and the number ‘e’.
Log to base 10 are known as common log
Log to base e are called natural logarithm
Use ‘ln’ to indicate ‘loge’
The rules of logarithm also apply to natural logarithm i.e<br>
slide13. Natural Logarithm (cont) Example 7: Use the properties of logarithm to simplify the following expressions<br>
slide14. Change of Base A logarithm of x with base a can be changed to base b and the formula is given as follows:<br>
slide15. Solving equations involving indices Example 8: Given that 2.142x = 4.78, find the values of x, correct to four decimal places.
Example 9: Solve 0.49x+1 = 0.62x.<br>
slide16. Solve logarithm equations Example 10: Given that
find the values of x.
Example 11: Solve<br>
slide2. Definition of a logarithm If a number y can be written in the form of ax, the index x is called the logarithm of y to the base a.
In other word, if y = ax, then logay = x where y and a are positive numbers.
Example 1: Write the exponent 3x = 16 into the logarithm form
Example 2: Write the exponent y-5 = 4.6 into the logarithm form<br>
slide3. Definition of a logarithm (cont) Example 3: Find the value of x in each of the following equations
(a) log3x = 4
(b) log2x = -5
(c) log10(x – 4) = 3<br>
slide4. Properties of Logarithm From the definition of logarithm, we can proceed further to obtain the following properties
(a) loga1 = 0
(b) loga a = 1
(c) loga an = n<br>
slide5. Properties of Logarithm (cont) Example 4: Without using the calculator, evaluate the following.
log101
log2.5(5/2)
log2 25
log381
log40.25
log8 2<br>
slide6. The Rules of Logarithm Proof:
Let loga x = m and loga y = n.
So, x = am and y = an.
Thus, xy = amï‚´ an = a(m+n)
Applying the definition of a logarithm gives<br>
slide7. The Rules of Logarithm (cont) Proof:
Let loga x = m and loga y = n.
So, x = am and y = an.
Thus,
Applying the definition of a logarithm gives<br>
slide8. The Rules of Logarithm (cont) Proof:
Let loga x = m with x = am.
Raising each side to the power r gives
Applying the definition of a logarithm gives<br>
slide9. The Rules of Logarithm (cont) Example 5: Simplify the following as a single logarithm<br>
slide10. The Rules of Logarithm (cont) Example 6: Without using a calculator, evaluate the following.<br>
slide11. The Rules of Logarithm (cont)<br>
slide12. Natural Logarithm The most frequently used bases for log are 10 and the number ‘e’.
Log to base 10 are known as common log
Log to base e are called natural logarithm
Use ‘ln’ to indicate ‘loge’
The rules of logarithm also apply to natural logarithm i.e<br>
slide13. Natural Logarithm (cont) Example 7: Use the properties of logarithm to simplify the following expressions<br>
slide14. Change of Base A logarithm of x with base a can be changed to base b and the formula is given as follows:<br>
slide15. Solving equations involving indices Example 8: Given that 2.142x = 4.78, find the values of x, correct to four decimal places.
Example 9: Solve 0.49x+1 = 0.62x.<br>
slide16. Solve logarithm equations Example 10: Given that
find the values of x.
Example 11: Solve<br>