1
1.1
Exponential to and Logarithmic Expressions
Exponential Equation to Logarithmic Equations
A logarithm is defined by y = bx if and only if logb y = x. This “if and only if”
often represented in math equations by the double ended arrow, “ ⇐⇒ ”, means
that if one statement is true the other is true as well. In the earlier sections,
the exponential was graphed, and from that graph, it inverse, the logarithm was
graphed. For example, if y = 2x and x = 3, then x = 3 is the input and 23 = 8
is the output. The logarithm reverses this process. It begins with the 8 and
ends with the 3. This is written as log2 8 = 3.
The equation logb y = x is read, “the log in base b of y is x.” In the exponential
equation of y = 2x , the 2 was called the base, and it is the same base in the
equation log2 y = x.
Example. Write 34 = 81 in as a logarithm.
In this problem the 3 is the base. The 4 was the input and the 81 was the output.
These switch roles in the logarithm. We write 34 = 81
⇐⇒
log3 81 = 4.
Example. Write 103 = 1000 as a logarithm.
We know that 103 = 1000 ⇐⇒ log10 1000 = 3; however, this is not the final
version of the answer. When the base of the log is 10, the 10 is implied. This is
called the common log. Thus log10 1000 = 3 is written log 1000 = 3.
Example. Write e0 = 1 as a logarithm.
Then, e0 = 1 ⇐⇒ loge 1 = 0; however, this is not how the answer is usually
written. When the logarithm is in base e, this is called the natural log and is
represented by ln. Thus, loge 1 = 0 is written ln 1 = 0.
Practice Problems. Write the following exponential equations in logarithmic
form.
1
1. 53 = 125
2. 4 2 = 2
3. 103 = 1000
4. 2−1 =
5. 3−2 =
6. 10−4 = 0.0001
1
9
1
7. 6 3
9. 2x = 16
1
2
8. 101 = 10
√
10. 3x = 3
1
11.
a0 = 1
12.
ey = a
13.
e1 = e
14.
e0 = 1
15.
ee = y
16.
e−4
17.
e−1
18.
e−x
19.
e2x = y
20.
ey = π
1.2
Logarithmic Equations to Exponential Equations
The exponential and logarithmic equations are inverses. If one is true then the
other is true. As an exponential can be written as a logarithm, a logarithm can
be written as an exponential.
y = bx
⇐⇒
logb y = x
Example. Write log2 32 = 5 as an exponential equation.
In the equation log2 32 = 5, which is read, “The log in base 2 of 32 is 5 means
2 raised to what power is 32. The answer is 5. Thus,
log2 32 = 5
⇐⇒
25 = 32
The most assured item is that the base of the logarithm is also the base of the
exponential.
Example. Write log 0.001 = −3 in exponential form.
When the base is not written, it is implied to be 10. Thus,
log 0.001 = −3
log10 0.001 = −3
⇐⇒
10−3 = 0.001
Rember that the base of the logarithm will be the base of the exponential.
Example. Write ln π = x in exponential form.
For the natural log, the base is e. Thus,
ln π = x
loge π = x
⇐⇒
ex = π
Rember that the base of the logarithm will be the base of the exponential.
Practice Problems. Write the following logarithmic equations in exponential
form.
2
21. log2 8 = 3
√
23. log5 5 =
22. log3 81 = 4
1
2
24. log7
1
7
= −1
25. log 10 = 1
26. ln 3 = x
27. logb 1 = 0
28. log2 x = 5
29. log4 x = −2
30. log(x) = − 32
1.3
Evaluating Logarithms
In the problems examined thus far, the problems have been solved. The excercise
has been to understand that given one form, it can be written in another. In
this section logarithms will be evaluated, without a calculator, by converting to
exponential form, then solving the exponential as done earlier.
Example. Evaluate, or solve, log5 625. This problem could also be expressed
as, ‘Evaluate log5 625 =”, or “Solve log5 625 =?”, or, “Simplify log5 625 = x”.
Most often the expression will be written in the first presentation. If this seems
strange remember first learing arithmetic. For example, “Solve 2 + 3 = ”.
This was later written as, “Simplify 2 + 3 =.” Finally it was left as, “Simplify
2 + 3”. This is the same presentation but for logarithms.
Write log5 625 in exponential form. Thus, log5 625 = x
⇐⇒
5x = 625.
Notice that the left hand side is in base 5. The reader should take a moment
and try to write 625 in base 5. Then,
5x = 625
5x = 54 Bases match, so exponents must.
=⇒ x = 4
Thus log5 625 = 4.
Example. Simplify log2
√
8.
This problem challenges the reader to recall rational exponents.
√
Let log2 8 = x
⇐⇒
2x =
√
8
1
2
= 8 Write 8 in base 2.
3
1
= (23 ) 2
3
2x = 2 2
3
=⇒ x =
2
Practice Problems. Without the use of a calculator evaluate the following
logarithms. (Hint: write the expression in exponential form.)
31. log6 36 =?
32. log4 64 =?
33. log3 81 =?
34. log 1, 000, 000 =?
√
36. log7 7
35. log5 ( 51 )
√
37. log2 3 4
1.4
38. log(0.001)
Logarithms and a Calculator
Simplifying logarithms by hand is limited and can only be evaluated with the
arguement of the log can be expressed in the base of the logarithm. When this is
not possible, them a decimal approximation can be obtained using a calculator.
Calculators usually come with two log options: the natural log ln x, and the
common log log x. Thus the calculator is also limited in the calculations it can
can do. Consider the following problems. The reader should use a calculator
and make sure the same results are obtained. Calculators vary on how to input,
see your calculator’s instructions for assistance.
Example Use a calculator to find a decimal approximation of log 2.
From the calculator, log 2 = 0.30103.
Example Use a calculator to find a decimal approximation of ln 2.
From the calculator, ln 2 = 0.69315.
Example Use a calculator to find a decimal approximation of log3 2.
From the calculator, log3 2 = 0.63093; however, your calculator does not have a
log in base 3 option. Problems such as this will be the next problem to solve.
Practice Problems. Use a calculator to evaluate the following logarithms.
Round your answer to five decimal places.
4
39. log 20
40. log(1.67)
41. log(784)
42. log(0.06)
4
43. log( 327
)
44. log π
45. ln 2
46. ln 10
47. ln 58
48. ln( π1 )
1.5
Change of Base Formula
Practice Problems. Use a calculator and the change of base formula to evaluate the following logarithms. Round your answers to five decimal places.
Consider the problem above to evaluate log3 2. This cannot be done directly,
but it can be done indirectly using the change of base formula.
log x
, or
log b
ln x
y = logb x =
ln b
y = logb x =
Using the change of base formula the log3 2 can be evaluated.
y = log3 2
log 2
=
= 0.63093, or
log 3
ln 2
= 0.63093
=
ln 3
Notice that the change of base formula works using either the common log or
the natual log. Either obtains the correct answer. The reader should try both.
49. log2 (100)
50. log3 (79)
51. log5 0.4
52. log36
53. log 14 (5)
54. logπ (0.0001)
1
100
55. log100 e
5