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CHAPTER 2
LESSON 6
Teacher’s Guide
Using Logs to Evaluate Other Logs
AW 2.3
MP 2.7
Objectives:
• To evaluate logs with bases other than 10.
• To work with the change of base law.
Evaluating logs with bases other than 10
Strategy:
To evaluate any logarithm, always start by assigning an unknown variable such as “x” to
the logarithm itself.
Example 1:
Evaluate log 8 70 to 9 decimals accuracy.
G–213
Notice that your calculator does not have a “ base 8” logarithm key.
So, to evaluate log 8 70 , the first important step is to let x = log 8 70 .
Rewriting x = log 8 70 as an exponential equation, we have
8x = 70
The second important step is to take the base 10 logarithm of each side of the equation.
(This is a valid operation, since the logarithms of equal numbers are equal.)
log8 x = log70
Using the law of logarithms for exponents, we have
xlog8 = log70
Dividing both sides of the equation by log8 , we have the final result:
log70 ⋅
= 2.043094339 (to nine decimals accuracy)
x=
log8
Note that this answer makes sense, because 82.0000 = 64 .
Check: 82.043094339 = 70
The Change of Base Law.
Let’s look at the result we obtained in the last example:
x = log 8 70 =
log 70
log8
We can generalize to obtain the following result (using base 10 logs).
loga
logb
To evaluate logarithms with bases other than 10, it is a simple matter to use this formula.
log b a =
Example 2
Evaluate log 5 160 to 4 decimals accuracy.
log 5 160 =
log160 ⋅
= 3.1534
log5
In Example 1, we took the base 10 logarithms of both sides of the equation. Of course,
we are not restricted to base 10: we could have used any base (say base c) in this
operation. This gives us the following general formula, which we call the change of base
law.
log c a
log c b
a > 0, b > 0, c > 0, b ≠ 1, c ≠ 1
log b a =
Change of Base Law
Example 3:
This means that we can express any logarithm in an infinite number of ways, using an
infinite variety of different bases.
log7 log 9 7 log 5 7 log15 7 log b 7 ⋅
=
=
=
=
= 2.807354922
log 2 7 =
log2 log 9 2 log 5 2 log15 2 log b 2
where b is any positive real number.
Example 4:
Express
log 5 7
as a single logarithm.
log 5 2
Using the change of base law,
log 5 7
= log 2 7
log 5 2
Proof of the Change of Base Law
Let x = logb a
Then b x = a
Taking the base c log of both sides, we have
log cb x = log c a
Using the power law,
xlogc b = log c a
logc a
x = log b a =
logc b
Example 5:
Use your graphing calculator to sketch the curve y = log 3 x .
Since your graphing calculator does not have a base 3 log key, we need to use the change
of base law. Using the dots on the grid, sketch what you see on your calculator.
log x
y = log 3 x =
log 3
Extensions of the Change of Base Law
(Note to Teachers: The following formulas were introduced to Math 12 students on
the 1999-2000 Problem Set. Although these laws are not included in the texts, they
are useful in solving more complex log equations. See Lesson 8 (ex. 5) for an
example of a regular provincial exam question that is readily solved using Law 2
below.)
Law 2
log b n x n = logb x
n ∈ℜ, b > 0, b ≠ 1, x > 0
Example 6:
Convert log 3 y to a base 9 logarithm.
log 3 y
= log 3 2 y 2
= 2log 9 y
Proof of Law 2:
Let y = log b x
Then x = b y
Raising both sides of the equation to the power n, we can write
xn = (b y )n = (b n )y .
Therefore, by the definition of logarithm, we have
log b n x n = y = log b x .
Law 3
1
log a b
a > 0, a ≠ 1, b > 0, b ≠ 1
log b a =
Example 7:
Convert
1
to a base 25 logarithm.
log y 5
1
log y 5
= log 5 y
= log 5 2 y 2
= log 25 y 2
Proof of Law 3:
log b a
loga
=
logb
1
=
logb
loga
1
=
log a b