5. Let a, b,and c be _____
real numbers such that b + 1. The logarith
of a with base bis denoted by
and is defined as
if a = b
6. Logarithmic functions and exponential functions are_______
7. In logarithmic form logb a, b cannot be______
8. The base in the given logarithmic expression log3 5 is______
9. If the base is not written in the logarithmic expression, then it is understood to
be_____
10. From the given log, 343, it is the same as asking "What will be the exponent of
? Since 7- = 343Therefore, log7 343 =
then,
logy 343 =
to get_____?since 7____343 therefore, log7 343 =____,then log7 343 = _______
Answers
Answer:
order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is
10
x
=
500
10x=500 where x represents the difference in magnitudes on the Richter Scale. How would we solve for x?
We have not yet learned a method for solving exponential equations algebraically. None of the algebraic tools discussed so far is sufficient to solve
10
x
=
500
10x=500. We know that
10
2
=
100
102=100 and
10
3
=
1000
103=1000, so it is clear that x must be some value between 2 and 3 since
y
=
10
x
y=10x is increasing. We can examine a graph to better estimate the solution.
Graph of the intersections of the equations y=10^x and y=500.
Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph above passes the horizontal line test. The exponential function
y
=
b
x
y=bx is one-to-one, so its inverse,
x
=
b
y
x=by is also a function. As is the case with all inverse functions, we simply interchange x and y and solve for y to find the inverse function. To represent y as a function of x, we use a logarithmic function of the form
y
=
l
o
g
b
(
x
)
y=logb(x). The base b logarithm of a number is the exponent by which we must raise b to get that number.
We read a logarithmic expression as, “The logarithm with base b of x is equal to y,” or, simplified, “log base b of x is y.” We can also say, “b raised to the power of y is x,” because logs are exponents. For example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since
2
5
=
32
25=32, we can write
l
o
g
2
32
=
5
log232=5. We read this as “log base 2 of 32 is 5.”
We can express the relationship between logarithmic form and its corresponding exponential form as follows:
l
o
g
b
(
x
)
=
y
⇔
b
y
=
x
,
b
>
0
,
b
≠
1
logb(x)=y⇔by=x,b>0,b≠1
Note that the base b is always positive.