Question
What is the first step in establishing that
log
(
10
7
⋅
5
)
=
7
+
log
5
?(1 point)
Responses
The first step is to make 5 an exponent, such that
log
(
10
7
⋅
5
)
=
log
(
10
7
⋅
5
)
.
The first step is to make 5 an exponent, such that logarithm left parenthesis 10 superscript 7 baseline times 5 right parenthesis equals logarithm left parenthesis 10 superscript 7 times 5 baseline right parenthesis .
The first step is to move the exponent in front of the logarithm, such that
log
(
10
7
⋅
5
)
=
7
log
(
10
⋅
5
)
.
The first step is to move the exponent in front of the logarithm, such that logarithm left parenthesis 10 superscript 7 baseline times 5 right parenthesis equals 7 logarithm left parenthesis 10 times 5 right parenthesis .
The first step is to rewrite the expression using multiplication, such that
log
(
10
7
⋅
5
)
=
log
10
7
⋅
log
5
.
The first step is to rewrite the expression using multiplication, such that logarithm left parenthesis 10 superscript 7 baseline times 5 right parenthesis equals logarithm 10 superscript 7 baseline times logarithm 5 .
The first step is to rewrite the expression using addition, such that
log
(
10
7
⋅
5
)
=
log
10
7
+
log
5
.
log
(
10
7
⋅
5
)
=
7
+
log
5
?(1 point)
Responses
The first step is to make 5 an exponent, such that
log
(
10
7
⋅
5
)
=
log
(
10
7
⋅
5
)
.
The first step is to make 5 an exponent, such that logarithm left parenthesis 10 superscript 7 baseline times 5 right parenthesis equals logarithm left parenthesis 10 superscript 7 times 5 baseline right parenthesis .
The first step is to move the exponent in front of the logarithm, such that
log
(
10
7
⋅
5
)
=
7
log
(
10
⋅
5
)
.
The first step is to move the exponent in front of the logarithm, such that logarithm left parenthesis 10 superscript 7 baseline times 5 right parenthesis equals 7 logarithm left parenthesis 10 times 5 right parenthesis .
The first step is to rewrite the expression using multiplication, such that
log
(
10
7
⋅
5
)
=
log
10
7
⋅
log
5
.
The first step is to rewrite the expression using multiplication, such that logarithm left parenthesis 10 superscript 7 baseline times 5 right parenthesis equals logarithm 10 superscript 7 baseline times logarithm 5 .
The first step is to rewrite the expression using addition, such that
log
(
10
7
⋅
5
)
=
log
10
7
+
log
5
.
Answers
GPT-4o mini
The correct first step in establishing that
\[
\log(10^7 \cdot 5) = 7 + \log(5)
\]
is:
The first step is to rewrite the expression using addition, such that
\[
\log(10^7 \cdot 5) = \log(10^7) + \log(5).
\]
This follows from the logarithmic property that states \(\log(a \cdot b) = \log(a) + \log(b)\). Once this step is made, you can then evaluate \(\log(10^7)\) as \(7\) (because \(10^7\) is equivalent to \(10\) raised to the 7th power).
Thus, the final conclusion will be:
\[
\log(10^7 \cdot 5) = \log(10^7) + \log(5) = 7 + \log(5).
\]
\[
\log(10^7 \cdot 5) = 7 + \log(5)
\]
is:
The first step is to rewrite the expression using addition, such that
\[
\log(10^7 \cdot 5) = \log(10^7) + \log(5).
\]
This follows from the logarithmic property that states \(\log(a \cdot b) = \log(a) + \log(b)\). Once this step is made, you can then evaluate \(\log(10^7)\) as \(7\) (because \(10^7\) is equivalent to \(10\) raised to the 7th power).
Thus, the final conclusion will be:
\[
\log(10^7 \cdot 5) = \log(10^7) + \log(5) = 7 + \log(5).
\]