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Mathematics A
1. Pure Mathematics
1.6 Exponentials and Logarithms
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Cards (265)
What is the general form of an exponential function?
f
(
x
)
=
f(x) =
f
(
x
)
=
a
x
a^{x}
a
x
The base
a
a
a
of an exponential function is typically greater than 1
The value of a^{x}</latex> is always positive for any real
x
x
x
What is the domain of an exponential function?
(
−
∞
,
∞
)
( - \infty, \infty)
(
−
∞
,
∞
)
The range of an exponential function is all positive real numbers,
(
0
,
∞
)
(0, \infty)
(
0
,
∞
)
What is the value of
f
(
0
)
f(0)
f
(
0
)
for any exponential function?
1
An
exponential function
is increasing when a > 1</latex>
An exponential function is decreasing when
0
<
a
<
1
0 < a < 1
0
<
a
<
1
True
Match the property with the corresponding exponential function:
Base is 2 ↔️
f
(
x
)
=
f(x) =
f
(
x
)
=
2
x
2^{x}
2
x
Base is 3 ↔️
g
(
x
)
=
g(x) =
g
(
x
)
=
3
x
3^{x}
3
x
Arrange the real-world applications of exponential functions in order of their relevance:
1️⃣ Population growth
2️⃣ Compound interest
3️⃣ Radioactive decay
What is the general form of a logarithmic function?
y
=
y =
y
=
log
a
x
\log_{a}x
lo
g
a
x
The base of a logarithm,
a
a
a
, is typically greater than 1
The relationship between a logarithmic function and its exponential form is
a
y
=
a^{y} =
a
y
=
x
x
x
What is the domain of a logarithmic function?
(
0
,
∞
)
(0, \infty)
(
0
,
∞
)
The range of a logarithmic function is all real numbers,
(
−
∞
,
∞
)
( - \infty, \infty)
(
−
∞
,
∞
)
What is the value of
log
a
1
\log_{a}1
lo
g
a
1
for any base
a
a
a
?
0
log
a
a
=
\log_{a}a =
lo
g
a
a
=
1
1
1
for all bases
a
a
a
Match the base with the corresponding logarithmic and exponential function:
2 ↔️
log
2
x
\log_{2}x
lo
g
2
x
and
2
x
2^{x}
2
x
10 ↔️
log
10
x
\log_{10}x
lo
g
10
x
and
1
0
x
10^{x}
1
0
x
e ↔️
ln
x
\ln x
ln
x
and
e
x
e^{x}
e
x
Arrange the real-world applications of logarithmic functions in order of their relevance:
1️⃣ pH calculations in chemistry
2️⃣ Richter scale for earthquakes
3️⃣ Decibel scale for sound levels
What is the relationship between a logarithmic function and its exponential form?
a
y
=
a^{y} =
a
y
=
x
x
x
What is a logarithmic function defined as?
y
=
y =
y
=
log
a
x
\log_{a}x
lo
g
a
x
In the logarithmic function
y
=
y =
y
=
log
a
x
\log_{a}x
lo
g
a
x
, the variable
a
a
a
represents the base
The domain of a logarithmic function is all
positive
real numbers.
The range of a logarithmic function is all
real numbers
.
What is
log
a
1
\log_{a}1
lo
g
a
1
equal to for any base
a
a
a
?
0
0
0
The value of
log
a
a
\log_{a}a
lo
g
a
a
is always 1
Match the base of the logarithmic function with its corresponding exponential function:
2
2
2
↔️
2
x
2^{x}
2
x
10
10
10
↔️
1
0
x
10^{x}
1
0
x
\log_{a}a = 1</latex> for all bases
a
a
a
.
What is the exponential form of
log
2
x
\log_{2}x
lo
g
2
x
?
2
x
2^{x}
2
x
The logarithmic function with base
10
10
10
is written as
log
10
x
\log_{10}x
lo
g
10
x
, and its exponential form is 10^{x}</latex>.10
What is a logarithmic function defined as in its general form?
y
=
y =
y
=
log
a
x
\log_{a}x
lo
g
a
x
In the logarithmic function y = \log_{a}x</latex>,
a
a
a
is called the base
The domain of a logarithmic function is all
positive
real numbers.
For any base
a
a
a
,
log
a
1
\log_{a}1
lo
g
a
1
is equal to 0
What is the value of \log_{a}a</latex> for any base
a
a
a
?
1
1
1
Match the base with its corresponding logarithmic and exponential functions:
Base 2 ↔️
log
2
x
\log_{2}x
lo
g
2
x
and
2
x
2^{x}
2
x
Base 10 ↔️
log
10
x
\log_{10}x
lo
g
10
x
and
1
0
x
10^{x}
1
0
x
For all exponential functions,
f
(
0
)
=
f(0) =
f
(
0
)
=
1
1
1
.
An exponential function is increasing when its base
a
a
a
is greater than 1
What is the product rule for logarithms?
\log_{a}(xy) = \log_{a}x + \log_{a}y</latex>
What is the quotient rule for logarithms?
log
a
(
x
y
)
=
\log_{a}\left(\frac{x}{y}\right) =
lo
g
a
(
y
x
)
=
log
a
x
−
log
a
y
\log_{a}x - \log_{a}y
lo
g
a
x
−
lo
g
a
y
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