Appendix N: Derivation of the Logarithm Change of Base Formula
We set out to prove the logarithm change of base formula: logb x = loga x loga b. To do so we let y = logb x and apply these as exponents on the base.
MATHEMATICS 0110A CHANGE OF BASE Suppose that we have
Let y = logb a. Then we know that this means that by = a. We can take logarithms to base c
Change of Base
6.2 Properties of Logarithms
out the inverse relationship between these two change of base formulas. To change the base of Prove the Quotient Rule and Power Rule for Logarithms.
S&Z . & .
Lesson 5-2 - Using Properties and the Change of Base Formula
You can prove the Change of Base. Formula blog X x because exponents and logarithms are inverses. Take the log base a of both sides: log
Elementary Functions The logarithm as an inverse function
The positive constant b is called the base (of the logarithm.) Smith (SHSU) Let's call this the “change of base” equation or “change of base” property.
. Logarithms (slides to )
Introduction to Algorithms
I can prove this using the definition of big-Omega: This tells us that every positive power of the logarithm of n to the base b where b ¿ 1
cs lect fall notes
3.3 The logarithm as an inverse function
Let's call this the “change of base” equation. Example. Suppose we want to compute log2(17) but our calculator only allows us to use the natural logarithm ln.
Lecture Notes . Logarithms
CS 161 Lecture 3 - 1 Solving recurrences
n ≥ 2 and the base cases of the induction proof (which is not the same as Plugging the numbers into the recurrence formula
lecture
Logarithms Math 121 Calculus II
Proof. By the inverse of the Fundamental Theorem of Calculus since lnx is defined as an In particular
logs
Section 4.3 Logarithmic Functions
expression. Properties of Logs: Change of Base. Proof: Let . Rewriting as an exponential gives . Taking the log base c of both sides of this equation gives.
logarithms
Section 4.3 Logarithmic Functions 1
A population of 50 flies is expected to double every week, leading to a function of the form , where x represents the number of weeks that have passed. When will this population reach 500? Trying to solve this problem leads to: Dividing both sides by 50 to isolate the exponential While we have set up exponential models and used them to make predictions, you may have noticed that solving exponential equations has not yet been mentioned. The reason is simple: none of the algebraic tools discussed so far are sufficient to solve exponential equations. Consider the equation above. We know that and , so it is clear that x must be some value between 3 and 4 since is increasing. We could use technology to create a table of values or graph to bette r estimate the solution. From the graph, we could better estimate the solution to be around 3.3. This result is still fairly unsatisfactory, and since the exponential function is one-to-one, it would be great to have an inverse function. None of the f unctions we have already discussed would serve as an inverse function and so we must introduce a new function, named log as the inverse of an exponential function. Since exponential functions have different bases, we will define corresponding logarithms o f different bases as well.Logarithm
The logarithm (base b) function, written , is the inverse of the exponential function (base b), . Since the logarithm and exponential are inverses, it follows that:Properties of Logs: Inverse Properties
Recall from the definition of an inverse function that if , then . Applying this to the exponential and logarithmic functions, we can convert between a logarithmic equation and its equivalent exponential. x xf)2(50)(=500 50(2) x 10 2 x 102=x 82
3 =162 4 () 2 x gx= ()x b log x b ()xb x b =log xb x b log caf=)(acf= 1
Logarithm Equivalent to an Exponential
The statement is equivalent to the statement .
Alternatively, we could show this by starting with the exponential function, then taking the log base b of both sides, giving . Using the inverse property of logs, we see that Since log is a function, it is most correctly written as , using parentheses to denote function evaluation , just as we would with f(c). However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written as .Example 1
Write these exponential
equations as logarithmic equations: a) b) c) a) is equivalent to b) is equivalent to c) is equivalent toExample 2
Write these logarithmic equations as exponential equations: a) b) a) is equivalent to b) is equivalent toTry it Now
1. Write the exponential equation as a logarithmic equation.
cb a =ac b =)(log a cb= log( )lo g a bb cb= log( ) b ca= )(logc b c b log 823 =255 2
10000110
4 823 =3)8(log 2 255
2 =2)25(log 5 4
11010000
101log410000=
216log
6 =()29log 3216log
6 =66 2/1 ()29log 3 =93 2 1642
Section 4.3 Logarithmic Functions 3
By establishing the relationship between exponential and logarithmic functions, we can now solve basic logarithmic and exponential equations by rewriting.Example 3
Solve for x.
By rewriting this expression as an exponential, , so x = 16.Example 4
Solve for x.
By rewriting this expression as a logarithm, we get .While this does
define a solution, and an exact solution at that, you may find it somewhat unsatisfying since it is difficult to compare this expression to the decimal estimate we madeSection 4.3 Logarithmic Functions 1
A population of 50 flies is expected to double every week, leading to a function of the form , where x represents the number of weeks that have passed. When will this population reach 500? Trying to solve this problem leads to: Dividing both sides by 50 to isolate the exponential While we have set up exponential models and used them to make predictions, you may have noticed that solving exponential equations has not yet been mentioned. The reason is simple: none of the algebraic tools discussed so far are sufficient to solve exponential equations. Consider the equation above. We know that and , so it is clear that x must be some value between 3 and 4 since is increasing. We could use technology to create a table of values or graph to bette r estimate the solution. From the graph, we could better estimate the solution to be around 3.3. This result is still fairly unsatisfactory, and since the exponential function is one-to-one, it would be great to have an inverse function. None of the f unctions we have already discussed would serve as an inverse function and so we must introduce a new function, named log as the inverse of an exponential function. Since exponential functions have different bases, we will define corresponding logarithms o f different bases as well.Logarithm
The logarithm (base b) function, written , is the inverse of the exponential function (base b), . Since the logarithm and exponential are inverses, it follows that:Properties of Logs: Inverse Properties
Recall from the definition of an inverse function that if , then . Applying this to the exponential and logarithmic functions, we can convert between a logarithmic equation and its equivalent exponential. x xf)2(50)(=500 50(2) x 10 2 x 102=x 82
3 =162 4 () 2 x gx= ()x b log x b ()xb x b =log xb x b log caf=)(acf= 1
Logarithm Equivalent to an Exponential
The statement is equivalent to the statement .
Alternatively, we could show this by starting with the exponential function, then taking the log base b of both sides, giving . Using the inverse property of logs, we see that Since log is a function, it is most correctly written as , using parentheses to denote function evaluation , just as we would with f(c). However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written as .Example 1
Write these exponential
equations as logarithmic equations: a) b) c) a) is equivalent to b) is equivalent to c) is equivalent toExample 2
Write these logarithmic equations as exponential equations: a) b) a) is equivalent to b) is equivalent toTry it Now
1. Write the exponential equation as a logarithmic equation.
cb a =ac b =)(log a cb= log( )lo g a bb cb= log( ) b ca= )(logc b c b log 823 =255 2
10000110
4 823 =3)8(log 2 255
2 =2)25(log 5 4
11010000
101log410000=
216log
6 =()29log 3216log
6 =66 2/1 ()29log 3 =93 2 1642
Section 4.3 Logarithmic Functions 3
By establishing the relationship between exponential and logarithmic functions, we can now solve basic logarithmic and exponential equations by rewriting.Example 3
Solve for x.
By rewriting this expression as an exponential, , so x = 16.Example 4
Solve for x.
By rewriting this expression as a logarithm, we get .While this does
define a solution, and an exact solution at that, you may find it somewhat unsatisfying since it is difficult to compare this expression to the decimal estimate we made- log base change formula proof