Mathematics is one of the engrossing subject, still students try to move away from this wonderful subject. But, my dear friends now, onwards you don’t need to move away from this subject as I am here to help my kids, so that they can easily grasp the concepts of this subject. Now, I will teach you all the math topics. Grade IX is the most important subject of student’s career as it decides where they have to go after this. In this article we will talk about the very crucial topic of algebra taught to students of class IX. Grade IX is the starting class of higher studies and from this class subject become tough and complex. If one has intention then he/she can easily reach to its goal with little efforts. So set your target and start making all possible efforts to reach it. Ok, now, move to the topic that we have to study today, the Logarithmic rules and formulas. Logarithmic is the new topic introduced in Grade IX.
What do you understand by this term what did it mean? No idea, ok , I will tell what is this, Logarithms is an exponent to which constant is raised to obtain a given number. In other words, we can also define it as:
The logarithm of a number represents the quantity as a power to which a fixed number has to be raised to produce a given number. The fixed number is known as base. For example, the logarithm 10000 to base is 4, because 10000 is 10 to the raised power 4 that means:
10000 = 104 = 10 x 10 x 10 x 10.
One more general from we can define it as if x = cy then y is the logarithm of x to base b, and is written logc(x), so log10(10000) = 4.
x = cy is the same as y = logbx
In simple words logarithm is just an exponent. Kids don’t get confused, let’s specify it, the logarithm of a number x to base b is just is just the exponent you put onto b to make the result equal x.
Let’s have an example to understand this concept, as you know 62 = 36, is in exponent form (the power is 2) and this is the logarithm of 25 to base 5. In above paragraph I explained you how to write log function, log6(36) = 2.
Let’s see an example to understand the basic concept of logarithm.
Log381 = ? is the same as 3? = 81,
In the second equation if we calculate the value of unknown than we can easily the value of equation one.
blog bx = x,
We can read it as the logarithm of x in the base b is the exponent you put on b to determine the value of x as a result.
3x = 81,
x = 81/3,
x = 27,
Now, we can easily put the value of known in the equation 1,
Log381 = 27.
In this way we can easily determine the value of unknown in log.
In mathematics we deal with the two types of logarithms: common logarithm and the natural logarithm. The general definition of log is given as: logax = N means that aN = x.
The common logarithm is given as: logx = log10x. in common log a logarithm is written without a base.
In natural logarithm the logarithm function is written ln and it is given as: ln x = logex where, e’s value is approximately 2.718.
For the above statements let x, y, a and b all are positive and a ≠ 1 and b ≠1.
Each language has its well – defined set of rules and if you will not follow the rules properly then you will get wrong result. In the same way mathematics has its set of rules and its topic contains a proper method and rules that you have to follow to get the correct answer. As, in this article we are talking about logarithm so now talk about, Logarithmic rules.
The first rule is known as the inverse property rule that depicts:
logaax = x and a(logax) = x
Is the product rule that says that: loga(xy)= logax + logay
Now, move towards the example so that you can understand what I want to explain you in above theory. Like: expand log3(2x),
Whenever any problem related to expansion arises it means that you have one log expression with lots of stuff inside it and you have to use the log rules and functions to get the complicated answer. In the above question we have 2x inside the log problem. In this question 2x is in multiplication form so apply the multiplication or product rule to get the answer. When we apply product rule we get:
log3(2x) = log3(2) + log3(x). And this is the answer of the problem.
The third one is defined as the Quotient rule and according to it:
.
For illustration of the above property let’s take an example and solve it with this you came to know how to use division or quotient rule.
Simplify: log4(16/x).
In this you can easily find that we have to apply division rule as the logarithmic function contains (16/x) in division form.
According to division rule we get:
log4(16/x) = log4(16) - log4(x),
The first term on the right hand side can be simplified to exact form or value, by applying the basic rule of the log that we discussed in the definition of the log.
log4(16) = 2, so, the final answer is:
log4(16/x) = 2 - log4(x).
After you get the answer don’t forget to cross check it and see that you properly expand all the terms or not.
The fourth rule is known as the power rule that shows:
loga(xp) = p logax.
To understand this property properly let’s take an example and solve it. log5(x3)
The exponent inside the log can be taken out as a multiplier
log5(x3) = 3.log5(x) = 3log5(x)
This is the simple property and according to this rule we take the power in the front side of log as a multiplier. This is simple one but most commonly used property of log used in almost each and every problem.
The last one is defined as the change of base formula:
Example for this: log3[4(x - 5)2 / x4 (x -1)3],
In this we will solve the question using the above property. In the first step we proceed as:
log3[4(x - 5)2 / x4 (x -1)3] = log3[4(x- 5)2] – log3[x4(x -1)3],
=[ log3(4) + log3[(x -5)2)] - log3(x4) + log3[(x - 1)3)]
= log3(4) + log3[[(x - 5)2] - log3(x4) – log3 [(x - 1)3]
= log3(4) + 2 log3[ (x -5)] - 4log3 (x) - 3log3[(x - 1).
Thus, this is the final answer of the above problem.
While solving logarithmic problem you all must keep few things in mind so that you can solve the question properly without any problems without mistakes.
loga (x + y) ≠ logax + logay,
loga(x - y) ≠ logax – logay.
Now, let’s see one more example that students can understand the different type of problem that occurs in logarithms and as you know example is the best way to learn and understand different concepts rule or functions. So here is one more example for you and see the step involved while solving the problem and with this you can easily practice lots of questions,
log2 (8x4 / 5)
Solution:
Step 1: 5 is divided into the 8x4 , so first split the numerator and denominator by using subtraction, on doing this we get:
log2(8x4 / 5) = log2 (8x+) - log2(5),
Step 2: in the next step don’t take the exponents out; it is only on the x, not on the 8 and we can take only those exponents that are present on every term inside the log. So in this split the factors using the addition,
log2 (8x4 ) - log2(5)= log2(8) + log2(x4) - log2(5),
Step 3: the x has an exponent so take it in front of log as a multiplier,
log2(8) + 4log2(x) - log2(5),
Since the power of 2 is 8, I can simplify the first log to an exact value:
log2(8 ) + 4log2(x) - log2(5),
3 + 4 log2 (x ) - log2(5)
Now, each log contains single value so the expression is simplified and the answer of the problem is, 3 + 4 log2(x ) - log2(5).
This is all about logarithmic rule and logarithm formula and for detail switch to online help.
What do you understand by this term what did it mean? No idea, ok , I will tell what is this, Logarithms is an exponent to which constant is raised to obtain a given number. In other words, we can also define it as:
The logarithm of a number represents the quantity as a power to which a fixed number has to be raised to produce a given number. The fixed number is known as base. For example, the logarithm 10000 to base is 4, because 10000 is 10 to the raised power 4 that means:
10000 = 104 = 10 x 10 x 10 x 10.
One more general from we can define it as if x = cy then y is the logarithm of x to base b, and is written logc(x), so log10(10000) = 4.
x = cy is the same as y = logbx
In simple words logarithm is just an exponent. Kids don’t get confused, let’s specify it, the logarithm of a number x to base b is just is just the exponent you put onto b to make the result equal x.
Let’s have an example to understand this concept, as you know 62 = 36, is in exponent form (the power is 2) and this is the logarithm of 25 to base 5. In above paragraph I explained you how to write log function, log6(36) = 2.
Let’s see an example to understand the basic concept of logarithm.
Log381 = ? is the same as 3? = 81,
In the second equation if we calculate the value of unknown than we can easily the value of equation one.
blog bx = x,
We can read it as the logarithm of x in the base b is the exponent you put on b to determine the value of x as a result.
3x = 81,
x = 81/3,
x = 27,
Now, we can easily put the value of known in the equation 1,
Log381 = 27.
In this way we can easily determine the value of unknown in log.
In mathematics we deal with the two types of logarithms: common logarithm and the natural logarithm. The general definition of log is given as: logax = N means that aN = x.
The common logarithm is given as: logx = log10x. in common log a logarithm is written without a base.
In natural logarithm the logarithm function is written ln and it is given as: ln x = logex where, e’s value is approximately 2.718.
For the above statements let x, y, a and b all are positive and a ≠ 1 and b ≠1.
Each language has its well – defined set of rules and if you will not follow the rules properly then you will get wrong result. In the same way mathematics has its set of rules and its topic contains a proper method and rules that you have to follow to get the correct answer. As, in this article we are talking about logarithm so now talk about, Logarithmic rules.
The first rule is known as the inverse property rule that depicts:
logaax = x and a(logax) = x
Is the product rule that says that: loga(xy)= logax + logay
Now, move towards the example so that you can understand what I want to explain you in above theory. Like: expand log3(2x),
Whenever any problem related to expansion arises it means that you have one log expression with lots of stuff inside it and you have to use the log rules and functions to get the complicated answer. In the above question we have 2x inside the log problem. In this question 2x is in multiplication form so apply the multiplication or product rule to get the answer. When we apply product rule we get:
log3(2x) = log3(2) + log3(x). And this is the answer of the problem.
The third one is defined as the Quotient rule and according to it:
.Simplify: log4(16/x).
In this you can easily find that we have to apply division rule as the logarithmic function contains (16/x) in division form.
According to division rule we get:
log4(16/x) = log4(16) - log4(x),
The first term on the right hand side can be simplified to exact form or value, by applying the basic rule of the log that we discussed in the definition of the log.
log4(16) = 2, so, the final answer is:
log4(16/x) = 2 - log4(x).
After you get the answer don’t forget to cross check it and see that you properly expand all the terms or not.
The fourth rule is known as the power rule that shows:
loga(xp) = p logax.
To understand this property properly let’s take an example and solve it. log5(x3)
The exponent inside the log can be taken out as a multiplier
log5(x3) = 3.log5(x) = 3log5(x)
This is the simple property and according to this rule we take the power in the front side of log as a multiplier. This is simple one but most commonly used property of log used in almost each and every problem.
The last one is defined as the change of base formula:
Example for this: log3[4(x - 5)2 / x4 (x -1)3],
In this we will solve the question using the above property. In the first step we proceed as:
log3[4(x - 5)2 / x4 (x -1)3] = log3[4(x- 5)2] – log3[x4(x -1)3],
=[ log3(4) + log3[(x -5)2)] - log3(x4) + log3[(x - 1)3)]
= log3(4) + log3[[(x - 5)2] - log3(x4) – log3 [(x - 1)3]
= log3(4) + 2 log3[ (x -5)] - 4log3 (x) - 3log3[(x - 1).
Thus, this is the final answer of the above problem.
While solving logarithmic problem you all must keep few things in mind so that you can solve the question properly without any problems without mistakes.
loga (x + y) ≠ logax + logay,
loga(x - y) ≠ logax – logay.
Now, let’s see one more example that students can understand the different type of problem that occurs in logarithms and as you know example is the best way to learn and understand different concepts rule or functions. So here is one more example for you and see the step involved while solving the problem and with this you can easily practice lots of questions,
log2 (8x4 / 5)
Solution:
Step 1: 5 is divided into the 8x4 , so first split the numerator and denominator by using subtraction, on doing this we get:
log2(8x4 / 5) = log2 (8x+) - log2(5),
Step 2: in the next step don’t take the exponents out; it is only on the x, not on the 8 and we can take only those exponents that are present on every term inside the log. So in this split the factors using the addition,
log2 (8x4 ) - log2(5)= log2(8) + log2(x4) - log2(5),
Step 3: the x has an exponent so take it in front of log as a multiplier,
log2(8) + 4log2(x) - log2(5),
Since the power of 2 is 8, I can simplify the first log to an exact value:
log2(8 ) + 4log2(x) - log2(5),
3 + 4 log2 (x ) - log2(5)
Now, each log contains single value so the expression is simplified and the answer of the problem is, 3 + 4 log2(x ) - log2(5).
This is all about logarithmic rule and logarithm formula and for detail switch to online help.
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